  Using geomterothermodynamics (GTD), we investigate the phase transition of
black hole in a metric independent way. We show that for any black hole,
curvature scalar (of equilibrium state space geometry) is singular at the point
where specific heat diverges. Previously such a result could only be shown by
taking specific examples on a case by case basis. A different type of phase
transition, where inverse specific heat diverges, is also studied within this
framework. We show that in the latter case, metric (of equilibrium state space
geometry) is singular instead of curvature scalar. Since a metric singularity
may be a coordinate artifact, we propose that GTD indicates that it is the
singularity of specific heat and not inverse specific heat which indicates a
phase transition of black holes.
