Wasserstein convergence rates of increasingly concentrating probability measures

For $\ell\colon \mathbb{R}^d \to [0,\infty)$ we consider the sequence of probability measures $\left(\mu_n\right)_{n \in \mathbb{N}}$, where $\mu_n$ is determined by a density that is proportional to $\exp(-n\ell)$. We allow for infinitely many global minimal points of $\ell$, as long as they form a finite union of compact manifolds. In this scenario, we show estimates for the $p$-Wasserstein convergence of $\left(\mu_n\right)_{n \in \mathbb{N}}$ to its limit measure. Imposing regularity conditions we obtain a speed of convergence of $n^{-1/(2p)}$ and adding a further technical assumption, we can improve this to a $p$-independent rate of $1/2$ for all orders $p\in\mathbb{N}$ of the Wasserstein distance.