I looked into that at some point. The original paper says $N^{O(d^2)}$, where $d$ is the dimension and $N$ is the size of the input (sum of bit lengths of matrix entries in $A\overline x\le \overline b$). A followup paper by Barvinok says that more careful analysis improves it down to $N^{O(d)}$, but it seems there is a mistake in the calculation. According to Barvinok himself (personal communication), the correct estimate is $N^{O(d \log d)}$. This is the bound we used in this paper (I. Pak, G. Panova, "On the complexity of computing Kronecker coefficients"), see Thm. 2.3.