  Computation of topological charges of the Schwarzschild and charged black
holes in AdS in canonical and grand canonical ensembles allows for a
classification of the phase transition points via the Bragg-Williams off-shell
free energy. We attempt a topological classification of the critical points and
the equilibrium phases of the dual gauge theory via a phenomenological matrix
model, which captures the features of the ${\cal{N}}=4$, $SU(N)$ Super
Yang-Mills theory on $S^3$ at finite temperature at large $N$. With minimal
modification of parameters, critical points of the matrix model at finite
chemical potential can be classified as well. The topological charges of
locally stable and unstable dynamical phases of the system turn out to be
opposite to each other, totalling to zero, and this matches the analysis in the
bulk.
