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Eschatology is a term used in Christian theology discussing "last things". However, "physical eschatology" does not have any religious connotations, just borrowing the word.
There are two interpretations of the phrase. The first is just a discussion of the "last things" of the universe. Theories about how the universe will change in the remote future, and the physical processes that will cause the universe to eventually end, perhaps in the big crunch, or due to The Heat Death of the Universe.
It is also used in transhumanist circles, with a different meaning. There it is considered a theoretical applied science, focused on how intelligent life can both affect and alter the very remote future, and how it can even survive in those times.
How far ahead do we mean by remote future? Well, the universe is about 1010 years old. Estimates as to when stars will cease to form due to material being used up in stellar processes is at about 1015 years - the "degenerate era". At about 1037 years, all elemental matter will finally have decayed due to proton decay (which is only theory at this point), and the "black hole era" begins, as black holes continue to evaporate. This will continue to about 10100 years, or the "dark era", where only neutrinos, electrons, positrons, and photons are left, and entropy has just about maximized. THAT's what we mean by remote future. Of course, this all assumes that the laws of physics don't change over time (evidence is that they don't), and that the relevant laws of physics are already known.
For a comparison, if there is enough matter in the universe for it to be closed, then the universe could collapse upon itself in as soon as 1010.8 years, much earlier than when heat death starts taking hold.
Freeman Dyson was one of the first people to approach the subject with concern to life, in his paper Time Without End: Physics and Biology in an Open Universe, published in (1976). An interesting feature in that paper is that one section discusses that if life were able to alter itself to an energy form, and use hibernation to allow it to alter subjective time over the long eternity of the "heat death" of the universe, it may be possible for the human race, at the complexity they are now, to save a finite amount of energy to use in this period, yet still exist subjectively forever into the future.