Emergent universe in Horava-Lifshitz-like F(R) gravity

We investigate the Einstein static universe (ESU) and the emergent universe scenario in the framework of Ho\v{r}ava-Lifshitz-like $F(R)$ gravity. We first perform a dynamical analysis in the phase space, and amongst others we show that a spatially open universe filled with matter satisfying the strong energy condition can exhibit a stable static phase. Additionally, we examine the behavior of the scenario under scalar perturbations and extract the conditions under which it is free of perturbative instabilities, showing that the obtained background ESU solutions are free of such instabilities. However, in order for the Einstein static universe to give rise to the emergent universe scenario we need to have an exotic matter sector that can lead the universe to depart from the stable static state and enter into its usual expanding thermal history.