  Following Cao-Hamilton-Ilmanen, in this paper we study the linear stability
of Perelman's $\nu$-entropy on Einstein manifolds with positive Ricci
curvature. We observe the equivalence between the linear stability restricted
to the transversal traceless symmetric 2-tensors and the stability of Einstein
manifolds with respect to the Hilbert action. As a main application, we give a
full classification of linear stability of the $\nu$-entropy on symmetric
spaces of compact type. In particular, we exhibit many more linearly stable and
linearly unstable examples than previously known and also the first linearly
stable examples, other than the standard spheres, whose second variations are
negative definite.
