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 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 [0, ) and converges to a geodesic sphere exponentially as t in the smooth topology. A key step is to show the L¹ 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.