Thermodynamics of Exotic Black Holes, Negative Temperature, and Bekenstein-Hawking Entropy

Recently, exotic black holes whose masses and angular momenta are interchanged have been found, and it is known that their entropies depend only on the $inner$ horizon areas. But a basic problem of these entropies is that the second law of thermodynamics is not guaranteed, in contrast to the Bekenstein-Hawking entropy. Here, I find that there is another entropy formula which recovers the usual Bekenstein-Hawking form, but the characteristic angular velocity and temperature are identified with those of the inner horizon, in order to satisfy the first law of black hole thermodynamics. The temperature has a $negative$ value, due to an upper bound of mass as in spin systems, and the angular velocity has a $lower$ bound. I show that one can obtain the same entropy formula from a conformal field theory computation, based on classical Virasoro algebras. I also describe several unanswered problems and some proposals for how these might be addressed.