  We investigate the Gauss-Bonnet dark energy model and its deformed version on
Ho\v{r}ava-Lifshitz cosmology, which belongs to the class of cosmologies
obtained from the so-called projectable version of Ho\v{r}ava-Lifshitz gravity.
In particular, we investigate the bulk/boundary interaction in this scenario
through the $Q$ function, which we interpret as a measure of the energy
transference between the bulk and the spacetime boundary. Then we discuss
whether the thermal equilibrium will be stable or not, once it is reached, and
the validity of the generalized second law. We show that the $Q$ function can
exhibit sign changes along the cosmic evolution and the Universe reaches the
thermal equilibrium as a transient phenomenon.
