  This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the
hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature
$\kappa$ evolving in direction of its normal vector, where the speed is given
by a power $\beta (\geq 1/m)$ of the $m$th mean curvature plus a volume
preserving 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 hypersurface
satisfies that the ratio of the biggest and smallest principal curvature is
close enough to 1 everywhere, depending only on $n$, $m$, $\beta$ and $\kappa$,
then under the flow this is maintained, there exists a unique, smooth solution
of the flow for all times, and the evolving hypersurfaces exponentially
converge to a geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$, enclosing the
same volume as the initial hypersurface.
