The 2nd Law of Thermodynamics argument is an intriguing one, but I'm not sure it works. Funny things happen on the way to infinity.

If you have an infinite state then you also have an inifitiely well ordered ground state from which at an infinitely long period of time ago you started decaying. But as we know, entropy depends combinatorically on states. I'm not up on my mathematical varieties of infinities, but I do know that a measurement that involves arranging particles (that's basically an entropy measurement) is much larger than some aleph-null or aleph-one set. I'm not sure how much bigger and I don't think many mathematicians (sadly) frequent this forum. Someone might try asking over at Straight Dope, but I suspect the answer will be nebulous there too.

My gut instinct is that there are so many different combinations to go through that you don't expect to live in a heat death universe if the universe has infinite extent in space and in time. If it is just inifite in time and not in space then you have a heat death. That's just a hunch though and I could easily be proven wrong.