Thermodynamics of the apparent horizon in infrared modified Horava-Lifshitz gravity

It is well known that by applying the first law of thermodynamics to the apparent horizon of a Friedmann-Robertson-Walker Universe and, assuming the entropy associated with the apparent horizon has the same form as the entropy formula for the static spherically symmetric black holes, one can derive the corresponding Friedmann equations in Einstein, Gauss-Bonnet, and more general Lovelock gravity. Is this a generic feature of any gravitational theory? Is the prescription applicable to other gravities? In this paper we would like to address the above questions by examining the same procedure for Horava-Lifshitz gravity. We find that in Horava-Lifshitz gravity, this approach does not work and we fail to reproduce a corresponding Friedmann equation in this theory by applying the first law of thermodynamics on the apparent horizon, together with the appropriate expression for the entropy in Horava-Lifshitz gravity. The reason for this failure seems to be due to the fact that Horava-Lifshitz gravity is not diffeomorphism invariant, and thus, the corresponding field equation cannot be derived from the first law around horizon in the spacetime. Without this, it implies that the specific gravitational theory is not consistent, which shows an additional problematic feature of Horava-Lifshitz gravity. Nevertheless, if we still take the area formula of geometric entropy and regard the Horava-Lifshitz sector in the Friedmann equation as an effective dark radiation, we are able to extract the corresponding Friedmann equation from the first law of thermodynamics. DOI: 10.1103/PhysRevD.87.024022