We investigate the convergence rate of the optimal entropic cost vε to the optimal transport cost as the noise parameter ε↓0. We show that for a large class of cost functions c on Rd×Rd (for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported and L᪲ marginals, one has vε − v0 = d/2 εlog(1/ε) + O(ε). Upper bounds are obtained by a block approximation strategy and an integral variant of Alexandrov's theorem. Under an infinitesimal twist condition on c, i.e. invertibility of ∇²xy c, we get the lower bound by establishing a quadratic detachment of the duality gap in d dimensions thanks to Minty's trick.