  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.
