Than Kyubey was totally wasting his time.
Recent work has cast some doubt on the heat death hypothesis and the applicability of any simple thermodynamic model to the universe in general.
Although entropy does increase in the model of an expanding universe, the maximum possible entropy rises much more rapidly, moving the universe further from the heat death with time, not closer.
I can't pretend to understand how this entropy bizz works (the creators of this show barely understand themselves) but if what this says is correct, then all I can say is "HAAAAAAAAAAAAAAAAAAAAH".
It's amazing how ten minutes on Wikipedia can teach you more about something than an entire anime based on it.