  In this paper, we study various aspects of the equilibrium thermodynamic
state space geometry of AdS black holes. We first examine the
Reissner-Nordstrom-AdS (RN-AdS) and the Kerr-AdS black holes. In this context,
the state space scalar curvature of these black holes is analysed in various
regions of their thermodynamic parameter space. This provides important new
insights into the structure and significance of the scalar curvature. We
further investigate critical phenomena, and the behaviour of the scalar
curvature near criticality, for KN-AdS black holes in two mixed ensembles,
introduced and elucidated in our earlier work arXiv:1002.2538 [hep-th]. The
critical exponents are identical to those in the RN-AdS and Kerr-AdS cases in
the canonical ensemble. This suggests an universality in the scaling behaviour
near critical points of AdS black holes. Our results further highlight
qualitative differences in the thermodynamic state space geometry for electric
charge and angular momentum fluctuations of these.
