  Black holes in Lorentz-violating theories have been claimed to violate the
second law of thermodynamics by perpetual motion energy extraction. We revisit
this question for a Penrose splitting process in a spherically symmetric
setting with two species of particles that move on radial geodesics that extend
to infinity. We show that energy extraction by this process cannot happen in
any theory in which gravity is attractive, in the sense of a geometric
inequality that we describe. This inequality is satisfied by all known
Einstein-\ae{}ther and Ho\v{r}ava black hole solutions.
