Let $(X,d)$ be a Polish metric space, let $\mathcal B(X)$ be its Borel $\sigma$-algebra, let $\rho$ be a Borel probability measure on $(X,\mathcal B(X))$, let $p\in\{1,2\}$, and let $C\in(0,\infty)$. Assume that $\rho$ satisfies the transport-entropy inequality $T_p(C)$, meaning that for every Borel probability measure $\nu$ on $(X,\mathcal B(X))$,

\begin{align*} W_p(\nu,\rho)^p\le C H(\nu\mid\rho), \end{align*}

where $H(\nu\mid\rho)$ denotes the relative entropy of $\nu$ with respect to $\rho$.

Let $V:X\to\mathbb R$ be a bounded Borel measurable function. Define

\begin{align*} Z:=\int_X e^{-V(x)}\,d\rho(x) \end{align*}

and define the Borel probability measure $\mu$ on $(X,\mathcal B(X))$ by

\begin{align*} \frac{d\mu}{d\rho}(x)=Z^{-1}e^{-V(x)} \end{align*}

for $\rho$-almost every $x\in X$. Let

\begin{align*} \operatorname{osc}(V):=\sup_{x\in X}V(x)-\inf_{x\in X}V(x). \end{align*}

Then $\mu$ satisfies $T_p(Ce^{\operatorname{osc}(V)})$; that is, for every Borel probability measure $\nu$ on $(X,\mathcal B(X))$,

\begin{align*} W_p(\nu,\mu)^p\le Ce^{\operatorname{osc}(V)}H(\nu\mid\mu). \end{align*}