Volume Entropy and Scalar Curvature near a Hyperbolic Metric

Let $(M^n,g_0)$ be a closed hyperbolic manifold, $n\ge 3$. We compute the Hessian of volume entropy at $g_0$ under the constraint $S{c}_g=-n(n-1)$ and express it as a quadratic form on transverse-traceless symmetric two-tensors. In dimension three, the Hessian is negative definite, and every sufficiently $C^{\mathrm{\infty }}$-close metric with $S{c}_g\ge -6$ has entropy at most $2$, with equality precisely for pullbacks of $g_0$. In dimensions $n\ge 4$, a universal spectral threshold determines the Hessian sign. Nonzero trace-free Codazzi tensors give positive directions, and hyperbolic bending provides smooth constrained entropy saddles in every such dimension.