  We develop a renormalization-group formalism for non-renormalizable theories
and apply it to Einstein gravity theory coupled to a scalar field with the
Lagrangian $L=\sqrt{g} [R U(\phi)-{1/2} G(\phi) g^{\mu\nu} \partial_{\mu}\phi
\partial_{\nu}\phi- V(\phi)]$, where $U(\phi), G(\phi)$ and $V(\phi)$ are
arbitrary functions of the scalar field. We calculate the one-loop counterterms
of this theory and obtain a system of renormalization-group equations in
partial derivatives for the functions $U, G$ and $V$ playing the role of
generalized charges which substitute for the usual charges in multicharge
theories. In the limit of a large but slowly varying scalar field and small
spacetime curvature this system gives the asymptotic behaviour of the
generalized charges compatible with the conventional choice of these functions
in quantum cosmological applications. It also demonstrates in the
over-Planckian domain the existence of the Weyl-invariant phase of gravity
theory asymptotically free in gravitational and cosmological constants.
