  In this paper, we propose a new definition of stable $(r+1)$-th capillary
hypersurfaces from variational perspective for any $1\leq r\leq n-1$. More
precisely, we define stable $(r+1)$-th capillary hypersurfaces to be smooth
local minimizers of a new energy functional under volume-preserving and contact
angle-preserving variations. Using the new concept of the stable $(r+1)$-th
capillary hypersurfaces, we generalize the stability results of Souam
\cite{Souam} in an Euclidean half-space and Guo-Wang-Xia \cite{GWX} in a
horoball in hyperbolic space for capillary hypersurface to $(r+1)$-th capillary
hypersurface case.
