  It is well known that by applying the first law of thermodynamics to the
apparent horizon of a Friedmann-Robertson-Walker universe, 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 Horrava-Lifshitz
gravity. Nevertheless, if we still take the area formula of geometric entropy
and regard 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.
