  For ensembles of Hamiltonians that fall under the Dyson classification of
random matrices with $\beta \in \{1,2,4\}$, the low-temperature mean entropy
can be shown to vanish as $\langle S(T)\rangle\sim \kappa T^{\beta+1}$. A
similar relation holds for Altland-Zirnbauer ensembles. JT gravity has been
shown to be dual to the double-scaling limit of a $\beta =2$ ensemble, with a
classical eigenvalue density $\propto e^{S_0}\sqrt{E}$ when $0 < E \ll 1$. We
use universal results about the distribution of the smallest eigenvalues in
such ensembles to calculate $\kappa$ up to corrections that we argue are doubly
exponentially small in $S_0$.
