We study Sinkhorn's algorithm for solving the entropically regularized optimal transport problem. Its iterate π_t is shown to satisfy H(π_t|π∗)+H(π∗|π_t)=O(1/t) where H denotes relative entropy and π∗ the optimal coupling. This holds for a large class of cost functions and marginals, including quadratic cost with subgaussian marginals. We also obtain the rate O(1/t) for the dual suboptimality and O(1/t^2) for the marginal entropies. More precisely, we derive non-asymptotic bounds, and in contrast to previous results on linear convergence that are limited to bounded costs, our estimates do not deteriorate exponentially with the regularization parameter. We also obtain a stability result for π∗ as a function of the marginals, quantified in relative entropy.