  We study the convergence of the transport plans $\gamma_\epsilon$ towards
$\gamma_0$ as well as the cost of the entropy-regularized optimal transport
$(c,\gamma_\epsilon)$ towards $(c,\gamma_0)$ as the regularization parameter
$\epsilon$ vanishes in the setting of finite entropy marginals. We show that
under the assumption of infinitesimally twisted cost and compactly supported
marginals the distance $W_2(\gamma_\epsilon,\gamma_0)$ is asymptotically
greater than $C\sqrt{\epsilon}$ and the suboptimality
$(c,\gamma_\epsilon)-(c,\gamma_0)$ is of order $\epsilon$. In the quadratic
cost case the compactness assumption is relaxed into a moment of order
$2+\delta$ assumption. Moreover, in the case of a Lipschitz transport map for
the non-regularized problem, the distance $W_2(\gamma_\epsilon,\gamma_0)$
converges to $0$ at rate $\sqrt{\epsilon}$. Finally, if in addition the
marginals have finite Fisher information, we prove
$(c,\gamma_\epsilon)-(c,\gamma_0) \sim d\epsilon/2$ and we provide a companion
expansion of $H(\gamma_\epsilon)$. These results are achieved by disentangling
the role of the cost and the entropy in the regularized problem.
