  We study a Boltzmann's type entropy functional (which appeared in existing
literature) defined on K\"ahler metrics of a fixed K\"ahler class. The critical
points of this functional are gradient K\"ahler-Ricci solitons, and the
functional was known to be monotonically increasing along the K\"ahler-Ricci
flow in the canonical class.
  In this article, we derive and analyze the second variation formula for this
entropy functional, and show that all gradient K\"ahler-Ricci solitons are
stable with respect to this entropy functional. Furthermore, using this result,
we give a new proof that gradient shrinking K\"ahler-Ricci solitons are stable
with respect to the Perelman's entropy in a fixed K\"ahler class.
