Phase Transition in JT Gravity and $T\bar{T}$ Deformation

In this paper we study a black hole phase transition in a generalized JT gravity noticed in arXiv:2006.03494. We investigate the effect of the phase transition on the Euclidean geodesic and holographic two-point function in models with dilaton potential which interpolates two ordinary JT gravities with different cosmological constants. It is noted that there exists a closed geodesic with a new scale at low temperature phase when the potential has a locally negative region. This scale causes several peaks in the two-point function. We also comment on the phase transition of charged black holes. We then consider coupling generalized JT gravity to a matter and study its relation to a $T\bar{T}$ deformation of CFT at the classical level. We find the deformation parameter as a function of the dilaton and provide examples showing Janus-type couplings.