Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 16; Problem P16.8.
The entropy of a black hole is given by
Given that the entopy of a thermal reservoir with temperature and internal energy (another fact from thermodynamics) is
where is a constant. If a black hole with mass is exchanging thermal radiation with the reservoir, we can say that the combined (and conserved) total energy of the system is so
Also from thermodynamics, it is known that when a system is in thermal equilibrium (no net transfer of radiation between the black hole and the reservoir), then the entropy should be at a local maximum. In this case, we can find the extrema of as a function of , since the mass of the black hole varies as it exchanges radiation with the reservoir. We get
However, from the second derivative:
we see that the entropy is actually at a local minimum at this mass and, since the curve 3 is a parabola, the maxima of entropy occur at the two ends and . Which of these two points is the overall maximum depends on and , but it does seem to indicate that equilibrium is achieved only when either the black hole evaporates and sends all its energy (mass) into the reservoir, or else the black hole swallows up the reservoir completely.