  We study the duality between JT gravity and the double-scaled matrix model
including their respective deformations. For these deformed theories we relate
the thermal partition function to the generating function of topological
gravity correlators that are determined as solutions to the KdV hierarchy. We
specialise to those deformations of JT gravity coupled to a gas of defects,
which conforms with known results in the literature. We express the
(asymptotic) thermal partition functions in a low temperature limit, in which
non-perturbative corrections are suppressed and the thermal partition function
becomes exact. In this limit we demonstrate that there is a Hawking-Page phase
transition between connected and disconnected surfaces for this instance of JT
gravity with a transition temperature affected by the presence of defects.
Furthermore, the calculated spectral form factors show the qualitative
behaviour expected for a Hawking-Page phase transition. The considered
deformations cause the ramp to be shifted along the real time axis. Finally, we
comment on recent results related to conical Weil-Petersson volumes and the
analytic continuation to two-dimensional de Sitter space.
