  We employ the Arnowitt-Deser-Misner formalism to study the renormalization
group flow of gravity minimally coupled to an arbitrary number of scalar,
vector, and Dirac fields. The decomposition of the gravitational degrees of
freedom into a lapse function, shift vector, and spatial metric equips
spacetime with a preferred (Euclidean) "time"-direction. In this work, we
provide a detailed derivation of the renormalization group flow of Newton's
constant and the cosmological constant on a flat Friedmann-Robertson-Walker
background. Adding matter fields, it is shown that their contribution to the
flow is the same as in the covariant formulation and can be captured by two
parameters $d_g$, $d_\lambda$. We classify the resulting fixed point structure
as a function of these parameters finding that the existence of non-Gaussian
renormalization group fixed points is rather generic. In particular the matter
content of the standard model and its most common extensions gives rise to one
non-Gaussian fixed point with real critical exponents suitable for Asymptotic
Safety. Moreover, we find non-Gaussian fixed points for any number of scalar
matter fields, making the scenario attractive for cosmological model building.
