The medians for exponential families and the normal law

Let $P$ a probability on the real line generating a natural exponential family $(P_t{)}_{t\in \mathbb{R}}$. We show that $t$ is a median of $P_t$ for all $t$ only if $P$ is the standard Gaussian law $N(0.1)\mathrm{.}$ The proof is based on the Choquet Deny equation.