  We prove a central limit theorem for the entropic transportation cost between
subgaussian probability measures, centered at the population cost. This is the
first result which allows for asymptotically valid inference for entropic
optimal transport between measures which are not necessarily discrete. In the
compactly supported case, we complement these results with new, faster,
convergence rates for the expected entropic transportation cost between
empirical measures. Our proof is based on strengthening convergence results for
dual solutions to the entropic optimal transport problem.
