  In this paper, we consider prescribed shifted Gauss curvature equations for
horo-convex hypersurfaces in $\mathbb{H}^{n+1}$. Under some sufficient
condition, we obtain an existence result by the standard degree theory based on
the a prior estimates for the solutions to the equations. Different from the
prescribed Weingarten curvature problem in space forms, we do not impose a sign
condition for radial derivative of the functions in the right-hand side of the
equations to prove the existence due to the horo-covexity of hypersurfaces in
$\mathbb{H}^{n+1}$.
