  In this paper, we study a mean curvature type flow with capillary boundary in
a horoball in hyperbolic space. Our flow preserves the volume of the bounded
domain enclosed by the hypersurface and monotonically decreases the energy
functional. We show that it has the long time existence and converges to a
truncated umbilical hypersurface in hyperbolic space. As an application, we
solve an isoperimetric type problem for hypersurfaces with capillary boundary
in a horoball.
