  Let $Q$ be a closed manifold with non-trivial first Betti number that admits
a non-trivial $S^1$-action, and $\Sigma \subseteq T^*Q$ a non-degenerate
starshaped hypersurface. We prove that the number of geometrically distinct
Reeb orbits of period at most $T$ on $\Sigma$ grows at least logarithmically in
$T$.
