  The quenched free energy, $F_Q(T){=}{-}T\langle \ln Z(T)\rangle$, of various
JT gravity and supergravity theories is explored, taking into account the key
non-perturbative physics that is accessible using their matrix model
formulations. The leading low energy physics of these systems can be modelled
by the Airy and (a family of) Bessel models, which arise from scaling limits of
matrix ensembles. The $F_Q(T)$s of these models are directly computed by
explicit sampling of the matrix ensembles, and how their properties are
connected to the statistical mechanics of the underlying discrete spectrum of
the ensembles is elucidated. Some of the low temperature ($T$) features of the
results confirm recent observations by Jassen and Mirbabayi. The results are
then used as benchmarks for exploring an intriguing formula proposed by Okuyama
for computing $F_Q(T)$ in terms of the connected correlators of its partition
function, the wormholes of the gravity theory. A low $T$ truncation of the
correlators helps render the formula practical, but it is shown that this is at
the expense of much of its accuracy. The significance of the statistical
interpretation of $F_Q(T)$ for black hole microphysics is discussed.
