In this paper we study the phases of a Schwarzschild black hole in the Anti
deSitter background geometry. Exploiting fluid/gravity duality we construct the
Maxwell equal area isotherm T=T* in the temperature-entropy plane, in order to
eliminate negative heat capacity black hole configurations. The construction we
present here is reminiscent of the isobar cut in the pressure-volume plane
which eliminates un-physical part of the Van der Walls curves below the
critical temperature. Our construction also modifies the Hawking-Page phase
transition. Stable black holes are formed at the temperature T > T*, while pure
radiation persists for T< T*. T* turns out to be below the standard
Hawking-Page temperature and there are no unstable black holes as in the usual
scenario. Also, we show that in order to reproduce the correct black hole
entropy S=A/4, one has to write a black hole equation of state, i.e. P=P(V), in
terms of the geometrical volume V=4\pi r^3/3.Comment: 15 pages, 4 Figures. Accepted for publication in Journal of Gravit
