What is the modern origin of the phrase "hell is the absence of God"?

1. St. Augustine said that the death of the soul (i.e., lack of sanctifying grace, as by not being baptized or by mortal sin) is the separation of the soul from God (Super Sent., lib. 2 d. 43 q. 1 a. 2 arg. 5: "mors animae est, ut Augustinus dicit separatio animae a Deo").

2. Those in hell lack sanctifying grace.

3. Therefore, those in hell are in a state of separation from God (i.e., they do not behold the beatific vision).

What is the modern origin of the phrase “hell is the absence of God”?

I do not believe that this phrase is a modern one. It has echoes that have come down to us throughout the centuries. But that said, I believe the modern phrase "hell is the absence of God" possibly became in a sense more common in popular culture due to the 2001 fantasy novelette by American writer Ted Chiang: "Hell Is the Absence of God" Ted Chiang certainly made this phrase popular.

The novelette is set in a world where the existence of God, souls, heaven, and hell are obvious and indisputable, and where miracles and angelic visitations are commonplace—albeit not necessarily benevolent. The wife of Neil, the main protagonist, is killed by the collateral damage of an angel's visitation. Knowing that his wife has ascended to heaven, the previously non-devout protagonist struggles to achieve the required love of God to join her.

The story also follows Janice, a woman born without legs who is made able-bodied in an angelic visitation, and Ethan, who cannot discern the meaning of an angelic encounter he experiences. - Hell Is the Absence of God

Prior to Ted Cheang's, Hell is the Absence of God, there appeared in February 1942 C.S. Lewis' The Screwtape Letters

This could possibly be another source of the popularity of the phrase in question in the 1950's - 1980's. Back in the 1980's this book was quite popular reading in Catholic high schools. I remember reading it in high school myself. The idea that God is not in hell comes across quite well in The Screwtape Letters.

The separation from God is pivotal to C. S. Lewis' vision of hell:

Separation seems for Lewis to describe the essential idea of hell, capturing what is conveyed by the biblical imagery of torture, destruction, and privation. To be forever cut off from God’s presence, eternally unable to know God’s love and mercy, would be a torture best described by being burned ceaselessly by fire. To be totally separated from other creatures, to be wholly and increasingly self-absorbed, makes that self smaller and smaller, and ultimately will result in the person ceasing to be a self. To someone who has been wholly centered on self, having that self cease to exist would be the ultimate possible loss, a horror describable for us, Lewis says, only through images of physical destruction. The torture of separation and the terror of ceasing to exist are better seen not as punishments imposed by God, but as the natural and inevitable outcome of choices humans themselves make and attitudes they themselves develop. - Heaven and Hell Idea and Image in C. S. Lewis

Nevertheless, I still feel the need to show that concept of "hell being the absence of God" in a more historical sense. There are echoes of it throughout the ages.

When I was a lad, I was taught how to make a perfect Act of Contrition, so I could pray it when I went to confession to a priest. Here is the one that I learned as a child:

O my God, I am heartily sorry for having offended Thee, and I detest all my sins, because I dread the loss of heaven, and the pains of hell; but most of all because they offend Thee, my God, Who are all good and deserving of all my love. I firmly resolve, with the help of Thy grace to confess my sins, to do penance and to amend my life. Amen. Perfect Contrition

One can see that I put the words the pains of hell in bold. This is done because one of the greatest pains one could possibly endure would be the fact that the damned are deprived of the presence of God.

”The pain of the damned,” says St. Thomas, ”is infinite, because it is the loss of an infinite good.” (1. 2, qu. 87, a. 4.) Such, too, is the doctrine of St. Bernard, who says, that the value of the loss of the damned is measured from the infinitude of God the supreme good.

Hence, hell does not consist in its devouring fire, nor in its intolerable stench, nor in the unceasing shrieks and bowlings of the damned, nor in the terrific sight of the devils, nor in the narrowness of that pit of torments, in which the damned are thrown one over the other: the pain which constitutes hell is the loss of God. In comparison of this pain, all the other torments of hell are trifling.

The reward of God’s faithful servants in heaven is, as he said to Abraham, God himself. ”I am thy reward, exceeding great.” (Gen. xv. 1.) Hence, as God is the reward of the blessed in heaven, so the loss of God is the punishment of the damned in hell.

Hence, St. Bruno has truly said, that how great soever the torments which may be inflicted on the damned, they never can equal the great pain of being deprived of God. Add torments to torments, but do not deprive them of God. ”Addantur tormenta tormentis, et Deo non priventur.” (Serm. de Jud. Fin.) According to St. Chrysostom, a thousand hells are not equal to this pain. Speaking of the loss of God, he said: ”Si mille dixeris gehennas, nihil par dices illius doloris.” (Hom, xlix., ad Pop.) God is so lovely that he deserves infinite love.

The sinner, drowned in sensual pleasures, scarcely knows God: he sees him only in the dark, and therefore he disregards the loss of God. But in hell he shall know God, and shall be tormented for ever by the thought of having voluntarily lost his infinite good. A certain Parisian doctor appeared after death to his bishop, and said that he was damned. His bishop asked him if he remembered the sciences in which he was so well versed in this life. He answered, that in hell the damned think only of the pain of having lost God.

”Depart from me, ye cursed, into everlasting fire. ” (Matt. xxv. 41.) “Depart from me.” This command constitutes the hell of the damned. Begone from me; you shall be no longer mine, and I shall be no longer yours. ”You are not my people, and I will not be yours.” (Osee i. 9.)

St. Augustine says, that if the damned saw the beauty of God, “they should feel no pain, and hell itself would be converted into a Paradise.” (Lib. de Trip. Hab.) But the damned shall never see God. When David forbade his son Absalom to appear in his presence, the sorrow of Absalom was so great, that he entreated Joab to tell his father that he would rather be put to death than never more be permitted to see his face. ”I beseech thee, therefore, that I may see the face of the king; and if he be mindful of my iniquity, let him kill me.” (2 Kings xiv. <32.) - ON THE PAIN OF LOSS WHICH THE DAMNED SUFFER IN HELL – St. Alphonsus

In hell there is no love. God is love. The damned hate and curse all the angels and saints. They curse particularly their guardian angels their special advocates and above all, the Virgin Mary, the Mother of Jesus. They hate the wounds of Jesus Christ, the blood of Jesus Christ, and the death of Jesus Christ. They hate the living of this world, especially those in the state of grace. Moreover they hate one another and themselves. How can God be present amongst such emptiness. The reprobates are thus left to themselves without God!

For those who are more into private revelation, here is something to ponder:

Our greatest torment consists in knowing with certainty that we will never see God. How greatly we are tortured by that which we were indifferent to while on earth! - Letter from a Soul in Hell

St. Augustine said, “The separation from God is a torment as great as God." Cf. Houdry, Bibliotheca concionatorum (Venice, 1786), vol 2, “Infernus,” No. 4, p. 427.