  Entropy bounds applied to a system of N species of light quantum fields in
thermal equilibrium at temperature T are saturated in four dimensions at a
maximal temperature T_max=M_Planck/N^1/2. We show that the correct setup for
understanding the reason for the saturation is a cosmological setup, and that a
possible explanation is the copious production of black holes at this maximal
temperature. The proposed explanation implies, if correct, that N light fields
cannot be in thermal equilibrium at temperatures T above T_max. However, we
have been unable to identify a concrete mechanism that is efficient and quick
enough to prevent the universe from exceeding this limiting temperature. The
same issues can be studied in the framework of AdS/CFT by using a brane moving
in a five dimensional AdS-Schwarzschild space to model a radiation dominated
universe. In this case we show that T_max is the temperature at which the brane
just reaches the horizon of the black hole, and that entropy bounds and the
generalized second law of thermodynamics seem to be violated when the brane
continues to fall into the black hole. We find, again, that the known physical
mechanisms, including black hole production, are not efficient enough to
prevent the brane from falling into the black hole. We propose several possible
explanations for the apparent violation of entropy bounds, but none is a
conclusive one.
