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We investigate properties of the pseudo-Riemannian volume, entropy, anddiameter for convex cocompact representations $\rho : \Gamma \to\mathrm{SO}(p,q+1)$ of closed $p$-manifold groups. In particular: We provide auniform lower bound of the product entropy times volume that depends only onthe geometry of the abstract group $\Gamma$. We prove that the entropy isbounded from above by $p-1$ with equality if and only if $\rho$ is conjugate toa representation inside ${\rm S}({\rm O}(p,1)\times{\rm O}(q))$, which answersaffirmatively to a question of Glorieux and Monclair. Lastly, we provefiniteness and compactness results for groups admitting convex cocompactrepresentations with bounded diameter.

Volume, entropy, and diameter in ${\rm SO}(p,q+1)$-higher Teichm\"uller spaces