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In the recent years the Schr\"odinger problem has gained a lot of attentionbecause of the connection, in the small-noise regime, with theMonge-Kantorovich optimal transport problem. Its optimal value, the\emph{entropic cost} $\mathscr{C}_T$, is here deeply investigated. In thispaper we study the regularity of $\mathscr{C}_T$ with respect to the parameter$T$ under a curvature condition and explicitly compute its first and secondderivative. As applications:  - we determine the large-time limit of $\mathscr{C}_T$ and provide sharpexponential convergence rates; we obtain this result not only for the classicalSchr\"odinger problem but also for the recently introduced Mean FieldSchr\"odinger problem [3];  - we improve the Taylor expansion of $T \mapsto T\mathscr{C}_T$ around $T=0$from the first to the second order.

A formula for the time derivative of the entropic cost and applications