  In this paper, we consider how to formulate semiclassical problems in the
context of the AdS/CFT correspondence, based on the proposal of Compere and
Marolf. Our prescription involves the effective action with self-action term
for boundary dynamical fields, which can be viewed as imposing mixed boundary
conditions for the gravity dual. We derive the semiclassical Einstein equations
sourced by boundary CFT stress-energy tensor. Analyzing perturbations of the
holographic semiclassical Einstein equations, we find a universal parameter
$\gamma_d$ which controls the contribution from boundary CFTs and specifies
dynamics on the AdS boundary. As a simple example, we examine the semiclassical
Einstein equations in $3$-dimensions with $4$-dimensional AdS gravity dual, and
show that the boundary BTZ black hole with vanishing expectation value of the
stress-energy tensor becomes unstable due to the backreaction from quantum
stress-energy tensor when the parameter $\gamma_d$ exceeds a certain critical
value.
