  We prove gradient estimates for hypersurfaces in the hyperbolic space
$\mathbb{H}^{n+1},$ expanding by negative powers of a certain class of
homogeneous curvature functions. We obtain optimal gradient estimates for
hypersurfaces evolving by certain powers $p>1$ of $F^{-1}$ and smooth
convergence of the properly rescaled hypersurfaces. In particular, the full
convergence result holds for the inverse Gauss curvature flow of surfaces
without any further pinching condition besides convexity of the initial
hypersurface.
