We study Sinkhorn's algorithm for solving the entropically regularized optimal transport problem. Its iterate $π_{t}$ is shown to satisfy $H(π_{t}|π_{*})+H(π_{*}|π_{t})=O(t^{-1})$ 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 $O(t^{-1})$ for the dual suboptimality and $O(t^{-2})$ 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.