  We establish weak limits for the empirical entropy regularized optimal
transport cost, the expectation of the empirical plan and the conditional
expectation. Our results require only uniform boundedness of the cost function
and no smoothness properties, thus emphasizing the far-reaching regularizing
nature of entropy penalization. To derive these results, we employ a novel
technique that sidesteps the intricacies linked to empirical process theory and
the control of suprema of function classes determined by the cost. Instead, we
perform a careful linearization analysis for entropic optimal transport with
respect to an empirical $L^2$-norm, which enables a streamlined analysis. As a
consequence, our work gives rise to new implications for a multitude of
transport-based applications under general costs, including pointwise
distributional limits for the empirical entropic optimal transport map
estimator, kernel methods as well as regularized colocalization curves.
Overall, our research lays the foundation for an expanded framework of
statistical inference with empirical entropic optimal transport.
