The Buddha and the Gods realm

I am sure I found this in one sutta unless my memory lapses for some reason. The following is the link to the list. You find a reference to some sources.

https://www.accesstoinsight.org/ptf/dhamma/sagga/loka.html

the gods realm is part of the cycle of existence.

Correct, while devas' lifespan is much longer than ours and they enjoy a lot more bliss/pleasure, they too will die eventually:

This was said by the Blessed One, said by the Arahant, so I have heard:

"When a deva is about to pass away from the company of devas, five omens appear: his garlands wither, his clothes get soiled, sweat comes out of his armpits, a dullness descends on his body, he no longer delights in his own deva-seat. The devas, knowing from this that 'This deva-son is about to pass away,' encourage him with three sayings: 'Go from here, honorable sir, to a good destination. Having gone to a good destination, gain the gain that is good to gain. Having gained the gain that is good to gain, become well-established.'" ~~ Iti 76 ~~

The fate of the gods does not always end well.

AN1.356-358:1.1: “… the sentient beings who die as gods and are reborn as humans are few, while those who die as gods and are reborn in hell, or the animal realm, or the ghost realm are many.”

AN1.353-355:1.1: “… the sentient beings who die as gods and are reborn as gods are few, while those who die as gods and are reborn in hell, or the animal realm, or the ghost realm are many.”

It might therefore be better as humans to let the gods be as they may and make good use of this very life to practice the Noble Eightfold Path.

Another reference is MN 49 in which the Buddha contradicts a God (named Baka Brahma):

When this was said, I told Baka Brahma, 'How immersed in ignorance is Baka Brahma! How immersed in ignorance is Baka Brahma! — in that what is actually inconstant he calls "constant." What is actually impermanent he calls "permanent." What is actually non-eternal he calls "eternal." What is actually partial he calls "total." What is actually subject to falling away he calls "not subject to falling away." Where one takes birth, ages, dies, falls away, and reappears, he says, "For here one does not take birth, does not age, does not die, does not fall away, does not reappear." And there being another, higher escape, he says, "There is no other, higher escape."

Here's one reference in DN 1:

"There comes a time, bhikkhus, when after the lapse of a long period this world contracts (disintegrates). While the world is contracting, beings for the most part are reborn in the Ābhassara Brahma-world. There they dwell, mind-made, feeding on rapture, self-luminous, moving through the air, abiding in glory. And they continue thus for a long, long period of time.

"But sooner or later, bhikkhus, after the lapse of a long period, there comes a time when this world begins to expand once again. While the world is expanding, an empty palace of Brahmā appears. Then a certain being, due to the exhaustion of his life-span or the exhaustion of his merit, passes away from the Ābhassara plane and re-arises in the empty palace of Brahmā. There he dwells, mind made, feeding on rapture, self-luminous, moving through the air, abiding in glory. And he continues thus for a long, long period of time.

"Then, as a result of dwelling there all alone for so long a time, there arises in him dissatisfaction and agitation, (and he yearns): 'Oh, that other beings might come to this place!' Just at that moment, due to the exhaustion of their life-span or the exhaustion of their merit, certain other beings pass away from the Ābhassara plane and re-arise in the palace of Brahmā, in companionship with him. There they dwell, mind-made, feeding on rapture, self-luminous, moving through the air, abiding in glory. And they continue thus for a long, long period of time.

"Thereupon the being who re-arose there first thinks to himself: 'I am Brahmā, the Great Brahmā, the Vanquisher, the Unvanquished, the Universal Seer, the Wielder of Power, the Lord, the Maker and Creator, the Supreme Being, the Ordainer, the Almighty, the Father of all that are and are to be. And these beings have been created by me. What is the reason? Because first I made the wish: "Oh, that other beings might come to this place!" And after I made this resolution, now these beings have come.'

"And the beings who re-arose there after him also think: 'This must be Brahmā, the Great Brahmā, the Vanquisher, the Unvanquished, the Universal Seer, the Wielder of Power, the Lord, the Maker and Creator, the Supreme Being, the Ordainer, the Almighty, the Father of all that are and are to be. And we have been created by him. What is the reason? Because we see that he was here first, and we appeared here after him.'

"Herein, bhikkhus, the being who re-arose there first possesses longer life, greater beauty, and greater authority than the beings who re-arose there after him.

"Now, bhikkhus, this comes to pass, that a certain being, after passing away from that plane, takes rebirth in this world. Having come to this world, he goes forth from home to homelessness. When he has gone forth, by means of ardor, endeavor, application, diligence, and right reflection, he attains to such a degree of mental concentration that with his mind thus concentrated he recollects his immediately preceding life, but none previous to that. He speaks thus: 'We were created by him, by Brahmā, the Great Brahmā, the Vanquisher, the Unvanquished, the Universal Seer, the Wielder of Power, the Lord, the Maker and Creator, the Supreme Being, the Ordainer, the Almighty, the Father of all that are and are to be. He is permanent, stable, eternal, not subject to change, and he will remain the same just like eternity itself. But we, who have been created by him and have come to this world, are impermanent, unstable, short-lived, doomed to perish.'