  We consider the probability measure minimizing a free energy functional equal
to the sum of a Coulomb interaction, a confinement potential and an entropy
term, which arises in the statistical mechanics of Coulomb gases. In the limit
where the inverse temperature $\beta$ tends to $\infty$ the entropy term
disappears and the measure, which we call the "thermal equilibrium measure"
tends to the well-known equilibrium measure, which can also be interpreted as a
solution to the classical obstacle problem. We provide quantitative estimates
on the convergence of the thermal equilibrium measure to the equilibrium
measure in strong norms in the bulk of the latter, with a sequence of explicit
correction terms in powers of $1/\beta$, as well as an analysis of the tail
after the boundary layer of size $\beta^{-1/2}$.
