Is there a term or phrase for the desire for Apocalypse?

I believe Sigmund Freud called this desire Thanatos. More commonly, it's called death instinct.

A primitive impulse for destruction, decay, and death, postulated by Sigmund Freud as coexisting with and opposing the life instinct. Also called Thanatos.

[American Heritage Dictionary]

Catastrophism

Wired's article 'This is Not The Apocalypse You Were Looking For' offers lots of ways of talking about those who seem to yearn for the end of the world.

For a single-word choice, there's catastrophism. This is defined more often as

The theory that changes in the earth's crust during geological history have resulted chiefly from sudden violent and unusual events.

But there is a second definition available in American English:

an outlook envisioning imminent catastrophe

For example: In contrast, the rest of the right offered "catastrophism": the conviction that the Republic could be overthrown only by violence.

Millenarianism

Originally Millenarinism was a wholly Christian concept, based on the 'Millenium' of the Book of Revelation, which foretells a 1000-year period of direct rule of humankind by God, after a great catastrophe.

Blessed and holy are those who share in the first resurrection. Over these the second death has no power, but they will be priests of God and of Christ, and they will reign with him a thousand years. - Revelation ch. 20

However the definition has widened to (as here in Merriam-Webster):

belief in a coming ideal society and especially one created by revolutionary action

As per Wikipedia:

Increasingly in the study of apocalyptic new religious movements, millenarianism is used to refer to a more cataclysmic and destructive arrival of a utopian period...