  We prove a new Minkowski type formula for capillary hypersurfaces supported
on totally geodesic hyperplanes in hyperbolic space. It leads to a
volume-preserving flow starting from a star-shaped initial hypersurface. We
prove the long-time existence of the flow and its uniform convergence to a
$\theta$-totally umbilical cap. Additionally, we establish that a
$\theta$-totally umbilical cap is an energy minimizer for a given enclosed
volume.
