  In this paper, we investigate the volume comparison theorem related to
$\sigma_2$-curvature. In particular, we show that volume comparison theorem
with respect to $\sigma_2$-curvature holds for metrics close to strictly stable
positive Einstein metrics. By applying similar techniques, we derive the local
rigidity theorem for strictly stable Ricci flat manifolds with respect to
$\sigma_2$-curvature, which shows it admits no metric with positive
$\sigma_2$-curvature near strictly stable Ricci-flat metrics.
