  Singularities in the thermodynamics of Kerr-Newman black holes are commonly
associated with phase transitions. However, such interpretations are
complicated by a lack of stability and, more significantly, by a lack of
conclusive insight from microscopic models. Here, I focus on the later problem.
I use the thermodynamic Riemannian curvature scalar $R$ as a try to get
microscopic information from the known thermodynamics. The hope is that this
could facilitate matching black hole thermodynamics to known models of
statistical mechanics. For the Kerr-Newman black hole, the sign of $R$ is
mostly positive, in contrast to that for ordinary thermodynamic models, where
$R$ is mostly negative. Cases with negative $R$ include most of the simple
critical point models. An exception is the Fermi gas, which has positive $R$. I
demonstrate several exact correspondences between the two-dimensional Fermi gas
and the extremal Kerr-Newman black hole. $R$ diverges to $+\infty$ along curves
of diverging heat capacities $C_{J,\Phi}$ and $C_{\Omega,Q}$, but not along the
Davies curve of diverging $C_{J,Q}$. Finding statistical mechanical models with
like behavior might yield additional insight into the microscopic properties of
black holes. I also discuss a possible physical interpretation of $|R|$.
