  Recently, the stochastic dynamical model based on the free energy landscape
was proposed to quantify the kinetics of the black hole phase transition. An
essential concept is the generalized free energy of the fluctuating black hole,
which was defined in terms of the thermodynamic relation previously. In this
work, by employing the Gibbons-Hawking path integral approach to black hole
thermodynamics, we show that the generalized free energy can be derived from
the Einstein-Hilbert action of the Euclidean gravitational instanton with the
conical singularity. This work provides a concrete and solid foundation for the
free energy landscape formalism of black hole phase transition.
