  We obtain the Ruppeiner geometry associated with the non-extended state space
($\Lambda$ constant) of the charged Gauss-Bonnet AdS (GB-AdS) black holes and
confirm that the state space Riemannian manifold becomes strongly curved in
regions where the black hole system develops strong statistical correlations in
the grand canonical ensemble ($M$ and $Q$ fluctuating). We establish the exact
proportionality between the state space scalar curvature $R$ and the inverse of
the singular free energy near the isolated critical point for the grand
canonical ensemble in spacetime dimension $d=5$, thus hopefully moving a step
closer to the agenda of a concrete physical interpretation of $R$ for black
holes. On the other hand, we show that while $R$ signals the Davies transition
points (which exist in GB-AdS black holes for $d \ge 6$) through its
divergence, it does not scale as the inverse of the singular free energy there.
Furthermore, adapting to the black hole case the ideas developed in
\cite{rupp2} in the context of pure fluids, we find that the state space
geometry encodes phase coexistence and first order transitions, identifies the
asymptotically critical region and even suggests a Widom line like crossover
regime in the supercritical region for $5-d$ case. The sign of $R$ appears to
imply a significant difference between the microscopic structure of the small
and the large black hole branches in $d=5$. We show that thermodynamic geometry
informs the microscopic nature of coexisting thermal GB-AdS and black hole
phases near the Hawking-Page phase transition.
