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The role of time and a possible foliation structure of spacetime are longstanding questions which lately received a lot of renewed attention from the quantum gravity community. In this talk, I will review recent progress in formulating a Wetterich-type functional renormalization group equation on foliated spacetimes and outline its potential applications. In particular, I will discuss first results concerning the RG flow of
Horava-Lifshitz gravity, highlighting a possible mechanism for a dynamical Lorentz-symmetry restoration at low energies.