Entropy-regularized optimal transport has become a key tool in optimal transport, providing both theoretical insight and efficient computation via the Sinkhorn algorithm. The regularization parameter plays the role of a temperature, and in this talk I will discuss the first-order Γ-expansion of the problem in the low-temperature limit. The starting point is to rewrite the regularized problem as a relative entropy with respect to a Gibbs measure whose energy is the duality gap. Using a mild extension of the Laplace method, we prove the convergence of these Gibbs measures as the temperature vanishes. We then derive a first-order Γ-expansion for non-degenerate ground costs, under suitable assumptions on the contact set and the marginals. This yields at the same time an entropic selection principle for the limit minimizer and an expansion of the regularized cost up to first order, generalizing existing results in the literature.