I think this came out pretty tidy:
If time has no beginning, then infinite time has passed.
If entropy increases monotonically with time (2nd law of thermo), then after infinite time it must either have diverged to infinity or converged to a maximum value.
Case 1: The heat death of the universe has not occurred, therefore entropy is not infinite.
Case 2: Internal combustion engines work, therefore entropy has not reached a maximum value.
Therefore time had a beginning. (modus tollens)
This still allows for appeals to very large or infinite volumes of space for life to arise, which then has to contend with the Fermi paradox.
Aeoli Pera 
Yes because our environment is still changing and unexpected (due to lack of heat death in our universe) we still experience novel things that dictate our lives. That is why infinite time does not exist.
>This still allows for appeals to very large or infinite volumes of space for life to arise, which then has to contend with the Fermi paradox.
Ah, but your argument fails if space is finite!
https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem
That’s right, I forgot we had this exact exchange years ago.
This may be the first time in my life I’ve felt I had a good reason to learn some higher math.
Hang on, finite and bounded volumes aren’t the same. This argument only works for a universe that looks completely different from this one (unless there’s some sort of parallel formalization?). Are you using your God-given brilliance to pull my leg?
Click to access dante.pdf
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