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In this paper we study the phases of a Schwarzschild black hole in the AntideSitter background geometry. Exploiting fluid/gravity duality we construct theMaxwell equal area isotherm T=T* in the temperature-entropy plane, in order toeliminate negative heat capacity black hole configurations. The construction wepresent here is reminiscent of the isobar cut in the pressure-volume planewhich eliminates un-physical part of the Van der Walls curves below thecritical temperature. Our construction also modifies the Hawking-Page phasetransition. Stable black holes are formed at the temperature T > T*, while pureradiation persists for T< T*. T* turns out to be below the standardHawking-Page temperature and there are no unstable black holes as in the usualscenario. Also, we show that in order to reproduce the correct black holeentropy S=A/4, one has to write a black hole equation of state, i.e. P=P(V), interms of the geometrical volume V=4\pi r^3/3.

Maxwell's equal area law and the Hawking-Page phase transition