We compute the renormalization group running of the Newton constant and the parameter $\ensuremath{\lambda}$ in ($3+1$)-dimensional projectable Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity. We use the background-field method, expanding around configurations with flat spatial metric, but nonvanishing shift. This allows us to reduce the number of interaction vertices and thereby drastically simplify the calculations. The gauge-invariant $\ensuremath{\beta}$-function of $\ensuremath{\lambda}$ has two families of zeros, attractive in the infrared and ultraviolet, respectively. They are candidates for the fixed points of the full renormalization group flow of the theory, once the $\ensuremath{\beta}$-functions for the rest of the couplings are added.