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In this article, we investigate the volume comparison with respect to scalarcurvature. In particular, we show volume comparison holds for small geodesicballs of metrics near a V-static metric. For closed manifold, we prove thevolume comparison for metrics near a strictly stable Einstein metric. Asapplications, we give a partial answer to a conjecture of Bray and recover aresult of Besson, Courtois and Gallot, which partially confirms a conjecture ofSchoen about closed hyperbolic manifold. Applying analogous techniques, weobtain a different proof of a local rigidity result due to Dai, Wang and Wei,which shows it admits no metric with positive scalar curvature near strictlystable Ricci-flat metrics.

Volume comparison with respect to scalar curvature