Stability of cosmological solutions in $F(R)$ Hořava-Lifshitz gravity

The present paper is devoted to the analysis of cosmological solutions and its stability in the frame of $F(R)$ Hořava-Lifshitz gravity. The perturbations around general spatially flat Friedmann-Lemaître-Robertson-Walker solutions are analyzed and it is shown that the stability of those solutions depends on the type of theory, i.e. on the form of the action $F(R)$, as well as on the extra parameters contained in every Hořava-Lifshitz theory (due to the breaking of Lorentz invariance). The (in)stability of cosmological solutions can provide a constraint of the models and it may give new observational predictions. A natural explanation of the end of inflation and radiation/matter phases can be provided by this class of theories. An explicit example of $F(R)$ gravity is studied, and the transition between the different epochs of the Universe history is achieved.