What's Wrong With Black Hole Thermodynamics?
Not only is the Bekenstein expression for the entropy of a black hole a convex function of the energy, rather than being a concave function as it must be, it predicts a final equilibrium temperature given by the harmonic mean. This violates the third law, and the principle of maximum work. The property that means are monotonically increasing functions of their argument underscores the error of transferring from temperature means to means in the internal energy when the energy is not a monotonically increasing function of temperature. Whereas the former leads to an increase in entropy, the latter lead to a decrease in entropy thereby violating the second law. The internal energy cannot increase at a slower rate than the temperature itself.