In this paper, we examine the volume comparison theorem associated with \(\sigma _k\) -curvature. Specifically, we demonstrate that the volume comparison theorem with respect to \(\sigma _k\) -curvature is valid for metrics closed to strictly stable, positive Einstein metrics. Utilizing analogous techniques, we derive a local rigidity theorem for strictly stable Ricci-flat manifolds with respect to \(\sigma _k\) -curvature. This theorem establishes that there are no metrics with positive \(\sigma _k\) -curvature in the vicinity of strictly stable Ricci-flat metrics.
