  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.
