  We study a volume/area preserving curvature flow of hypersurfaces that are
convex by horospheres in the hyperbolic space, with velocity given by a generic
positive, increasing function of the mean curvature, not necessarly
homogeneous. For this class of speeds we prove the exponential convergence to a
geodesic sphere. The proof is ispired by [10] and is based on the preserving of
the convexity by horospheres that allows to bound the inner and outer radii and
to give uniform bounds on the curvature by maximum principle arguments. In
order to deduce the exponential trend, we study the behaviour of a suitable
ratio associated to the hypersurface that converges exponentially in time to
the value associated to a geodesic sphere.
