Can ghost condensate decrease entropy?

By looking for possible violation of the generalized second law, we might be able to find regions in the space of theories and states that do not allow holographic dual descriptions. We revisit three proposals for violation of the generalized second law in the simplest Higgs phase of gravity called ghost condensate. Two of them, (i) analogue of Penrose process and (ii) semiclassical heat flow, are based on Lorentz breaking effects, by which particles of different species can have different limits of speed. We show that processes in both (i) and (ii) are always slower than accretion of ghost condensate and cannot decrease the total entropy before the accretion increases the entropy. The other proposal is to use (iii) negative energy carried by excitations of ghost condensate. We prove an averaged null energy condition, which we conjecture prevents the proposal (iii) from violating the generalized second law in a coarse-grained sense.