  In the AdS$_3$/CFT$_2$ framework, the Euclidean BTZ black hole corresponds to
the dominant high-temperature phase of its dual field theory. We initially
employ perturbative methods to solve the Einstein equations as boundary value
problems, providing correlators for the energy-momentum tensor operator at low
points. Utilizing operator equations established in our previous work, we
further compute arbitrary high-point correlators for the energy-momentum tensor
operator in the high-temperature phase and recursive relations for these
high-point functions. Concurrently, we employ the Chern-Simons formalism to
derive consistent results. Further, using the cut-off AdS/$T\bar{T}$-deformed
CFT duality, we calculate the energy-momentum tensor correlators, contributing
to the comprehensive understanding of the system's dynamics. Finally, stress
tensor correlators enable us to ascertain the corresponding KdV operator
correlators at low-temperature.
