  The black hole entropy has been observed to generically turn negative at
exponentially low temperatures $T\sim e^{-S_0}$ in the extremal
Bekenstein-Hawking entropy $S_0$, a seeming pathology often attributed to
missing non-perturbative effects. In fact, we show that this negativity must
happen for any effective theory of quantum gravity with an ensemble
description. To do so, we identify the usual gravitational entropy as an
annealed entropy $S_a$, and prove that this quantity gives $S_0$ at extremality
if and only if the ground-state energy is protected by supersymmetry, and
diverges negatively otherwise. The actual thermodynamically-behaved quantity is
the average or quenched entropy $S_q$, whose calculation is poorly understood
in gravity: it involves replica wormholes in a regime where the topological
expansion breaks down. Using matrix integrals we find new instanton saddles
that dominate gravitational correlators at $T\sim e^{-S_0}$ and are dual to
semiclassical wormholes involving dynamical branes. These brane solutions give
the leading contribution to any black hole very near extremality, and a duality
with matrix ensembles would not make sense without them. In the non-BPS case,
they are required to make $S_q$ non-negative and also enhance the negativity of
$S_a$, both effects consistent with matrix integrals evaluated exactly. Our
instanton results are tested against the on-shell action of D3-branes dual to
multiply wrapped Wilson loops in $\mathcal{N}=4$ super-YM, and a precise match
is found. Our analysis of low-energy random matrix spectra also explains the
origin of spectral gaps in supersymmetric theories, not only when there are BPS
states at zero energy, but also for purely non-BPS supermultiplets. In the
former, our prediction for the gap in terms of the degeneracy of BPS states
agrees with the R-charge scaling in gapped multiplets of $\mathcal{N}=2$
super-JT gravity.
