  We discuss the evolution of radiation and Bekenstein-Hawking entropies in
expanding isotropic universes. We establish a general relation which shows why
it is inevitable that there is currently a huge difference in the numerical
values of these two entropies. Some anthropic constraints on their values are
given and other aspects of the cosmological 'entropy gap' problem are
discussed. The coincidence of the classical and quantum entropies for black
holes with Hawking lifetime equal to the age of the universe, and hence of
radius equal to the proton size, is shown to be identical to the condition that
we observe the universe at the main-sequence lifetime.
