  We consider the volume preserving flow of smooth, closed and convex
hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1} (n\geq 2)$ with the
speed given by arbitrary positive power $\alpha$ of the Gauss curvature. We
prove that if the initial hypersurface is convex, then the smooth solution of
the flow remains convex and exists for all positive time $t\in [0,\infty)$.
Moreover, we apply a result of Kohlmann which characterises the geodesic ball
using the hyperbolic curvature measures and an argument of Alexandrov
reflection to prove that the flow converges to a geodesic sphere exponentially
in the smooth topology. This can be viewed as the first result for non-local
type volume preserving curvature flows for hypersurfaces in the hyperbolic
space with only convexity required on the initial data.
