In　this　paper,　we　examine　the　volume　comparison　theorem　associated　with　σk-curvature.　Specifically,　we　demonstrate　that　the　volume　comparison　theorem　with　respect　to　σk-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　σk-curvature.　This　theorem　establishes　that　there　are　no　metrics　with　positive　σk-curvature　in　the　vicinity　of　strictly　stable　Ricci-flat　metrics.　©　The　Author(s),　under　exclusive　licence　to　Springer-Verlag　GmbH　Germany,　part　of　Springer　Nature　2025.