But a de Sitter universe has no matter.
No, but I have plenty of extensions of de-Sitter that do...
While it certainly seems like we are becoming a de Sitter-style universe due to the expansion of the universe and the dominance of the cosmological constant, we aren't there now.
This isn't relevant. The point is that I have plenty of examples of past infinite space-times that do not reach any point of equilibrium.
But that said, if the universe were infinitely old, why haven't we achieved de Sitter status yet?
I'm not sure what you mean by "de-Sitter status". Yes, under certain behaviours of dark energy, the Universe will approximate de-Sitter behaviour, but that is not the point. I raised de-Sitter, as stressed above, simply to give an example of a past-infinite cosmology that does not approach equilibrium, not because our Universe maybe approaching the *future* state of de-Sitter. If we did live in a de-Sitter-type cosmology, which we probably don't based on evidence, then we would exist at some finite time following the de-Sitter "pinch", the pinch being what we think of as the Big Bang. The bulk of the infinite past would be the collapsing phase of de-Sitter, prior to the pinch.
If there's an infinite amount of energy to be had, where is it?
Distributed over a spatially infinite Universe. You just need to start thinking of densities rather than absolute quantities.
Classic Second Law. There is no perfect system. All processes bleed energy that can never be recovered.
Can never be recovered? There's that equilibrium again that doesn't necessarily exist. I would be careful about trying to use conventional concepts of thermodyanamics in a cosmological setting. Most need careful revision.
Except I've already stated that I do think that infinity does exist.
Yes, I appreciate that. That is why I stressed that my comment was relevant to the topic, rather than our own particular discussion.