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We compute the renormalization group running of the Newton constant and theparameter $\lambda$ in $(3+1)$-dimensional projectable Horava gravity. We usethe background field method expanding around configurations with flat spatialmetric, but non-vanishing shift. This allows us to reduce the number ofinteraction vertices and thereby drastically simplify the calculations. Thegauge invariant $\beta$-function of $\lambda$ has two families of zeros,attractive in the infrared and ultraviolet respectively. They are candidatesfor the fixed points of the full renormalization group flow of the theory, oncethe $\beta$-functions for the rest of the couplings are added.

Towards the renormalization group flow of Horava gravity in $(3+1)$ dimensions