Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space

We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space ${\mathbb{H}}^{n+1}$ with speed given by a general nonhomogeneous function of the Gauss curvature. For a large class of speed functions, we prove that the solution of the flow remains convex, exists for all positive time $t\in [0,\mathrm{\infty })$ and converges to a geodesic sphere exponentially as $t\to \mathrm{\infty }$ in the smooth topology. A key step is to show the $L^1$ oscillation decay of the Gauss curvature to its average along a subsequence of times going to the infinity, which combined with an argument using the hyperbolic curvature measure theory implies the Hausdorff convergence.