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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 firstvariational formula was obtained by Chang-Fang. We also prove that Einsteinmanifolds (with dimension $\ge 7$) with positive scalar curvature is a strictlocal maximum within its conformal class, unless the manifold is isometric toround sphere with the standard metric up to a multiple of constant. Note thatwhen $(M,g)$ is locally conformally flat, this functional reduces to thewell-studied $\int_M \sigma_3(g)dv_g$. Hence, our result generalize a previousresult of Jeff Viaclovsky without the locally conformally flat restraint.

The Second Variational Formula For the Functional $\int v^{(6)}(g)dV_g$