  In this paper, we study rigidity problems for hypersurfaces with constant
curvature quotients $\frac{\mathcal{H}_{2k+1}}{\mathcal{H}_{2k}}$ in the warped
product manifolds. Here $\mathcal{H}_{2k}$ is the $k$-th Gauss-Bonnet curvature
and $\mathcal{H}_{2k+1}$ arises from the first variation of the total
integration of $\mathcal{H}_{2k}$. Hence the quotients considered here are in
general different from $\frac{\sigma_{2k+1}}{\sigma_{2k}}$, where $\sigma_k$
are the usual mean curvatures. We prove several rigidity and Bernstein type
results for compact or non-compact hypersurfaces corresponding to such
quotients.
