  At the present paper, it is studied cosmological solutions and its stability
in the frame of F(R) Horava-Lifshitz gravity. The perturbations around general
spatially flat FRW solutions are analyzed and it is showed that the stability
of those solutions will depend on the kind of theory, i.e. on the form of the
action F(R), as well as on the parameters contained in any Horava-Lifshitz
theory due to the breaking of Lorentz invariance. The (in)stability of a given
cosmic solution can restrict the models and gives new observational
predictions, and can give a natural explanation on the end of inflation and
radiation/matter phases. An explicit example of F(R) is studied, and it is
showed that the instability can produce the transition between the different
epochs of the Universe history.
