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In this paper, we establish a new volume comparison theorem for a completemanifold with a function $\rho(x)$ as the lower bound of the Bakry-Emery Riccicurvature. As applications, we obtain a new volume rigidity result of thegradient Ricci almost solitons. Furthermore, we extend the results of Cao andZhou \cite{CZ} to shrinking gradient Ricci almost solitons and get the rigidityresult with respect to the maximal volume growth.

New Volume Comparison Results and Volume Growth Rigidity of Gradient Ricci Almost Solitons