  By using the Jacobi metric of the configuration space, and assuming
ergodicity, we calculate the Boltzmann entropy $S$ of a finite-dimensional
system around a non-degenerate critical point of its potential energy $V$. We
compare $S$ with the entropy of a quantum or thermal system with effective
potential $V_{eff}$. We examine conditions, up to first order in perturbation
theory, under which these entropies are equal.
