  In the recent years the Schr\"odinger problem has gained a lot of attention
because of the connection, in the small-noise regime, with the
Monge-Kantorovich optimal transport problem. Its optimal value, the
\emph{entropic cost} $\mathscr{C}_T$, is here deeply investigated. In this
paper we study the regularity of $\mathscr{C}_T$ with respect to the parameter
$T$ under a curvature condition and explicitly compute its first and second
derivative. As applications:
  - we determine the large-time limit of $\mathscr{C}_T$ and provide sharp
exponential convergence rates; we obtain this result not only for the classical
Schr\"odinger problem but also for the recently introduced Mean Field
Schr\"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.
