  We consider the quermassintegral preserving flow of closed \emph{h-convex}
hypersurfaces in hyperbolic space with the speed given by any positive power of
a smooth symmetric, strictly increasing, and homogeneous of degree one function
$f$ of the principal curvatures which is inverse concave and has dual $f_*$
approaching zero on the boundary of the positive cone. We prove that if the
initial hypersurface is \emph{h-convex}, then the solution of the flow becomes
strictly \emph{h-convex} for $t>0$, the flow exists for all time and converges
to a geodesic sphere exponentially in the smooth topology.
