This paper is devoted to the study of gradient \(\rho \) -Einstein soliton. We have derived volume growth estimate for non-trivial gradient shrinking \(\rho \) -Einstein soliton with non-negative scalar curvature. Later, an isometry theorem is obtained for non-trivial compact gradient \(\rho \) -Einstein soliton with a restriction on \(\rho \) . Further, it is showed under certain hypotheses that a compact gradient \(\rho \) -Einstein soliton is trivial. Finally, we established that the potential function in a compact gradient \(\rho \) -Einstein soliton is affine.
