How does Christ's hypostatic union affect the temporal nature of the Trinity?

Opposites cannot be said truly of the same thing in the same way: the divine and human things said of Christ are, of course, in opposition, suffering and incapable of suffering, for example, or dead and immortal

St. Thomas Aquinas - On the Trinity and the Incarnation Chapter 39

St. Thomas holds that Christs two natures are neither confused nor mixed. And Jesus' body, which has natural properties is subject to them. Christ's divine nature is not subject to the things that His human nature is subject to.

I think you're thinking that they're mixed somehow and Christ's human nature (and subsequent death) brings down the Trinity or splits it up for a bit. But that's just not contingent on the Incarnation. If so, the Trinity would have been affected by the Incarnation as well as by the Crucifixion and It would have been made more glorious by the Transfiguration and even more glorious by the Resurrection.

Furthermore,

It is clear also from this that, although the Son is incarnate, neither the Father nor the Holy Spirit, for all that, need be incarnate, since the Incarnation did not take place by a union in the nature in which the three divine Persons are together, but in hypostasis...

The Trinity is a plurality of Persons subsisting in one nature and the Incarnation is one Person subsisting in a plurality of natures.

So, Trinity and the Incarnations require more than complicated Venn Diagrams to explain.

  +-----------------+   (Incarnation)
 +                   +       | 
 +      Father    +--+-------|---+
 +        +------+DN +--+    |    +
  +------+-------+--+    + Son HN +
         + Spirit+       +   |    +
         +               +---|---+
          +-------------+    |

DN = divine Nature

HN = Human Nature

but an even better than a crummy ASCII venn diagram is every piece of religious art depicting Jesus ever

where you see Jesus' hand with two fingers and a thumb up representing the Blessed Trinity and two fingers down representing His two natures.

The first obvious point that should be established when discussing the “Hypostatic Union” is that this is one of the most mysterious doctrines of divine revelation, and must be understood as infinitely unattainable to the finite mind.

Our human existence is so saturated with temporal contingency that for finite human intellect to try to reconcile an eternal person’s existence with his temporal actuality is impossible without divine revelation. To put it simply – we will never fully understand the infinite existence of the Trinity – if we could we would be God.

Jesus is the Logos which is the eternal 2nd person of the Trinity. The Logos is existence by his very nature. Everything in the material universe from beginning to end was created by him via his eternal state of existence – including his own human nature. Although his divine nature existed from the beginning, (“In the beginning was the Word, and word was with God, and Word was God.”) his human nature unfolded in the temporal reality that is in our human nature out of material contingency.

As Aquinas puts it:

Since God became man "in the fullness oftime," the hypostatic union does not exist from eternity; itis the work or creation of God, and took place in time

The doctrine of Trinity that Christians have become accustomed to believe isn’t exactly formulated in Sacred Scripture as clearly as it has been in Sacred Tradition. The word Trinity cannot be found in the Bible. However, our understanding of this mystery has developed over time.

We can start by leaning on the Church Fathers to establish an intellectually clear foundation for unpacking this divinely revealed Truth:

God was in the beginning; but the beginning, we have been taught, is the power of the Logos. For the Lord of the universe, who is Himself the necessary ground of all being, inasmuch as no creature was yet in existence, was alone; but inasmuch as He was all power, Himself the necessary ground of things visible and invisible, with Him were all things; with Him, by Logos-power, the Logos Himself also, who was in Him, subsists. And by His simple will the Logos springs forth; and the Logos, not coming forth in vain, becomes the first-begotten work of the Father. Him (the Logos) we know to be the beginning of the world. But He came into being by participation, not by abscission; for what is cut off is separated from the original substance, but that which comes by participation, making its choice of function, does not render him deficient from whom it is taken. For just as from one torch many fires are lighted, but the light of the first torch is not lessened by the kindling of many torches, so the Logos, coming forth from the Logos-power of the Father, has not divested of the Logos-power Him who begat Him.
Tatian the Syrian, Oration Against the Greeks, 5 (c. A.D. 175).

We have already asserted that God made the world, and all which it contains, by His Word, and Reason, and Power. It is abundantly plain that your philosophers, too, regard the Logos--that is, the Word and Reason--as the Creator of the universe...And we, in like manner, hold that the Word, and Reason, and Power, by which we have said God made all, have spirit as their proper and essential substratum, in which the Word has inbeing to give forth utterances, and reason abides to dispose and arrange, and power is over all to execute. We have been taught that He proceeds forth from God, and in that procession He is generated; so that He is the Son of God, and is called God from unity of substance with God. For God, too, is a Spirit. Even when the ray is shot from the sun, it is still part of the parent mass; the sun will still be in the ray, because it is a ray of the sun--there is no division of substance, but merely an extension. Thus Christ is Spirit of Spirit, and God of God, as light of light is kindled. The material matrix remains entire and unimpaired, though you derive from it any number of shoots possessed of its qualities; so, too, that which has come forth out of God is at once God and the Son of God, and the two are one. In this way also, as He is Spirit of Spirit and God of God, He is made a second in manner of existence--in position, not in nature; and He did not withdraw from the original source, but went forth. This ray of God, then, as it was always foretold in ancient times, descending into a certain virgin, and made flesh in her womb, is in His birth God and man united.
Tertullian, Apology, 21 (A.D. 197).

The Logos alone of this God is from God himself; wherefore also the Logos is God, being the substance of God."
Hippolytus, Refutation against All Heresies, 10:29 (A.D. 220).

But if the Son is said to be sent by the Father on this account, that the one is the Father, and the other the Son, this does not in any manner hinder us from believing the Son to be equal, and consubstantial, and co-eternal with the Father, and yet to have been sent as Son by the Father. Not because the one is greater, the other less; but because the one is Father, the other Son; the one begetter, the other begotten; the one, He from whom He is who is sent; the other, He who is from Him who sends.
Augustine, On the Trinity, 4:20 (A.D. 416).

The Word of God, then, the only-begotten Son of the Father, in all things like and equal to the Father, God of God, Light of Light, Wisdom of Wisdom, Essence of Essence, is altogether that which the Father is, yet is not the Father, because the one is Son, the other is Father. And hence He knows all that the Father knows; but to Him to know, as to be, is from the Father, for to know and to be is there one. And therefore, as to be is not to the Father from the Son, so neither is to know. Accordingly, as though uttering Himself, the Father begat the Word equal to Himself in all things; for He would not have uttered Himself wholly and perfectly, if there were in His Word anything more or less than in Himself.
Augustine, On the Trinity, 15:14 (A.D. 416).

In the course of time, then, the Father forsooth was born, and the Father suffered, God Himself, the Lord Almighty, whom in their preaching they declare to be Jesus Christ. We, however, as we indeed always have done and more especially since we have been better instructed by the Paraclete, who leads men indeed into all truth), believe that there is one only God, but under the following dispensation, or oikonomia, as it is called, that this one only God has also a Son, His Word, who proceeded from Himself, by whom all things were made, and without whom nothing was made. Him we believe to have been sent by the Father into the Virgin, and to have been born of her--being both Man and God, the Son of Man and the Son of God, and to have been called by the name of Jesus Christ; we believe Him to have suffered, died, and been buried, according to the Scriptures, and, after He had been raised again by the Father and taken back to heaven, to be sitting at the right hand of the Father, and that He will come to judge the quick and the dead; who sent also from heaven from the Father, according to His own promise, the Holy Ghost, the Paraclete, the sanctifier of the faith of those who believe in the Father, and in the Son, and in the Holy Ghost. That this rule of faith has come down to us from the beginning of the gospel, even before any of the older heretics, much more before Praxeas, a pretender of yesterday, will be apparent both from the lateness of date which marks all heresies, and also from the absolutely novel character of our new-fangled Praxeas.
Tertullian, Against Praxeas, 2 (post A.D. 213).

Frank Sheed, who is considered the greatest Trinitarian theologian of the last century, does an incredible job of presenting the doctrine of the Trinity in light of all the great doctors of the Church.

He writes in his book “A Map of Life:”

Christ is God-made-man: that is He is truly God and He is truly man. He is God–with the nature of God: He took to Himself and made His own a complete human nature–a real human body and a real human soul. He is, then, one person–God–with two natures–divine and human. Nor is all this mere abstract matter, of no real concern to us. Everything in our life is bound up with the one person and the two natures of Christ. We must grasp this central luminous fact, or everything remains in darkness. The distinction between person and nature is not some deep and hidden thing to which philosophy only comes after centuries of study. It is, on the contrary, a distinction so obvious that the smallest child who can talk at all makes it automatically. If in the half-light he sees a vague outline that might be anything, he asks "What is that?" If, on the other hand, he can see that it is a human being, but cannot distinguish or does not recognize the features, he asks "Who is that?" The distinction between what and who is the distinction between nature and person. Of every man the two questions–what is he? and who is he?–can be answered. Every man, in other words, is both a nature and a person. Into my every action, nature and person enter. For instance I speak. I, the person, speak. But I am able to speak only because I am a man, because it is of my nature to speak. I discover that there are all sorts of things I can do: and all sorts of things I cannot do. My nature decides. I can think, speak, walk: these actions go with the nature of man, which I have. I cannot fly, for this goes with the nature of a bird, which I have not. My nature, then, decides what I can do: it may be thought of as settling the sphere of action possible to me. According to my nature, I can act: apart from it, I cannot. But my nature does not do these things–I, the person, do them. It is not my nature that speaks, walks, thinks: it is I, the person. A man may then be thought of as a person–who acts–and a nature–which decides the field in which he acts. In man there is simply one nature to one person. In Christ there are two natures to one person: and our minds used to the one-nature-to-one-person state of man tend to cry out that there is a contradiction in the idea of two natures to one person. But once it has been grasped that "person" and "nature" are not identical in meaning: once it has been grasped that the person acts and the nature is that principle in him which decides his sphere of action, then we see that mysterious as Our Lord's person and nature may be, there is no contradiction. God the Son, the Second Person of the Blessed Trinity,[1] assumed–took to Himself–a human nature: made it His own: not simply as something which He could use as a convenient sphere to act in, but really as His own: just as our nature is our own. In us the relation of person and nature is such that not merely do we say "I have a human nature" (as we might say "I have an umbrella") but person and nature are so fused in one concrete reality that we say "I am a man." So God the Son can say not only "I am God with a human nature to act in" but in the most absolute fullness of meaning He can say "I am man." He does not simply act as man: He is man–as truly man as we. This one person has two spheres of action: Christ our Lord could act either in His nature as God or in His nature as man. Remember the principle stated a few paragraphs back, that it is not the nature that acts, but the person. Therefore, whether He was acting in His divine nature or in His human nature, it was always the person who acted: and there was only the one person–God. Then this is the position. Christ is God: therefore whatever Christ did, God did. When Christ acted in His divine nature (as when He raised the dead to life) it was God who did it: when Christ acted in His human nature (as when He was born, suffered and died) it was God who did it: God was born, God suffered, God died. For it is the person who acts: and Christ is God.

All this being stated, I think perhaps the best and simplest answer for your question, "How does Christ's hypostatic union affect the temporal nature of the Trinity," is," In the beginning was the Word, and the Word was with God, and the Word was God. He was with God in the beginning. Through him all things were made; without him nothing was made that has been made. In him was life, and that life was the light of all mankind. The light shines in the darkness, and the darkness has not overcome it.