  We consider the Gauss curvature type flow for uniformly convex hypersurfaces
in the hyperbolic space $\mathbb{H}^{n+1}\ (n\geqslant 2)$. We prove that if
the initial closed hypersurface is smooth and uniformly convex, then the smooth
solution exists for all positive time and converges smoothly and exponentially
to a geodesic sphere centered at the origin. The key step is to prove the upper
bound of Gauss curvature and that uniform convexity preserves along the flow.
As an application, we provide a new proof for an Alexandrov-Fenchel inequality
comparing $(n-2)$th quermassintegral and volume of convex domains in
$\mathbb{H}^{n+1}\ (n\geqslant 2)$.
