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This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in thehyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature$\kappa$ evolving in direction of its normal vector, where the speed is givenby a power $\beta (\geq 1/m)$ of the $m$th mean curvature plus a volumepreserving term, including the case of powers of the mean curvature and of the$\mbox{Gau\ss}$ curvature. The main result is that if the initial hypersurfacesatisfies that the ratio of the biggest and smallest principal curvature isclose enough to 1 everywhere, depending only on $n$, $m$, $\beta$ and $\kappa$,then under the flow this is maintained, there exists a unique, smooth solutionof the flow for all times, and the evolving hypersurfaces exponentiallyconverge to a geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$, enclosing thesame volume as the initial hypersurface.

Volume-Preserving flow by powers of the mth mean curvature in the hyperbolic space