Let $H:A \to \mathbb R$ be a continuous function defined on a compact subset $A\subset \mathbf{R}^n$. Then the Laplace principle shows that $$ \lim_{\theta\to \infty }\frac{1}{\theta}\log \int_A e^{-\theta H(x)}dx =-\min_{x\in A} H(x). $$
I was wondering with additional regularity assumptions on $H$, can we establish a convergence rate in $\theta$? Has such a problem been studied in the literature?