  In this note, we compute the second variational formula for the functional
$\int_M v^{(6)}(g)dv_g$, which was introduced by Graham-Juhl and the first
variational formula was obtained by Chang-Fang. We also prove that Einstein
manifolds (with dimension $\ge 7$) with positive scalar curvature is a strict
local maximum within its conformal class, unless the manifold is isometric to
round sphere with the standard metric up to a multiple of constant. Note that
when $(M,g)$ is locally conformally flat, this functional reduces to the
well-studied $\int_M \sigma_3(g)dv_g$. Hence, our result generalize a previous
result of Jeff Viaclovsky without the locally conformally flat restraint.
