  Sharpening the holographic correspondence of Jackiw-Teitelboim (JT) gravity
and its dual matrix model description at a finite radial cutoff $\lambda$
through the $T\overline{T}$ deformation is of interest. To proceed, we simplify
the problem by considering the Airy model and deform Airy correlators in the
same way as in $T\overline{T}$-deformed JT gravity. We use those correlators to
compute the annealed and quenched free energies for both $\lambda > 0$ and
$\lambda < 0$ from an integral representation of the replica trick. At the
leading order in $\lambda$ and low temperatures, we confirm that the genus-zero
quenched free energy monotonically decreases as a function of temperature when
perturbation theory is valid. We then study the all-genus quenched free energy
at low temperatures, where we discover and discuss subtleties due to
non-perturbative effects in the Airy model, as well as the contributions from
the non-perturbative branch under the $T\overline{T}$ deformation.
