Let $Γ'<Γ$ be two discrete groups acting properly by isometries on a Gromov-hyperbolic space $X$. We prove that their critical exponents coincide if and only if $Γ'$ is co-amenable in $Γ$, under the assumption that the action of $Γ$ on $X$ is strongly positively recurrent, i.e. has a growth gap at infinity. This generalizes all previously known results on this question, which required either $X$ to be the real hyperbolic space and $Γ$ geometrically finite, or $X$ Gromov hyperbolic and $Γ$ cocompact. This result is optimal: we provide several counterexamples when the action is not strongly positively recurrent.