Varying alpha and black hole entropy

Recently it has been suggested that an increase in the fine structure constant alpha with time would decrease the entropy of a Reissner-Nordstrom black hole, thereby violating the second law of thermodynamics. In this note we point out that, at least for a certain class of charged dilaton black holes related to string theory, the entropy does not change under adiabatic variations of alpha and one might expect it to increase for non-adiabatic changes.