Moment-sum-of-squares hierarchies of semidefinite programs can be used to approximate the volume of a given compact basic semialgebraic set $K$. The idea consists of approximating from above the indicator function of $K$ with a sequence of polynomials of increasing degree $d$, so that the integrals of these polynomials generate a convergence sequence of upper bounds on the volume of $K$. We show that the asymptotic rate of this convergence is at least $O(1 / \log \log d)$.