Let $M$ be a complete, noncompact constant mean curvature hypersurface of finite index in $\mathbb{R}^{n+1}$ . We show that if either $M$ has zero volume entropy, or zero total curvature entropy and $n \leq 5$, or has bounded curvature and is properly embedded, then $M$ is minimal. We obtain similar results in more general ambient manifolds. Moreover the article contains some results of independent interest, about the volume entropy and the bottom of the essential spectrum.