  We study the renormalization group flow of gravity coupled to scalar matter
using functional renormalization group techniques. The novel feature is the
inclusion of higher-derivative terms in the scalar propagator. Such terms give
rise to Ostrogradski ghosts which signal an instability of the system and are
therefore dangerous for the consistency of the theory. Since it is expected
that such terms are generated dynamically by the renormalization group flow
they provide a potential threat when constructing a theory of quantum gravity
based on Asymptotic Safety. Our work then establishes the following picture:
upon incorporating higher-derivative terms in the scalar propagator the flow of
the gravity-matter system possesses a fixed point structure suitable for
Asymptotic Safety. This structure includes an interacting renormalization group
fixed point where the Ostrogradski ghosts acquire an infinite mass and decouple
from the system. Tracing the flow towards the infrared it is found that there
is a subset of complete renormalization group trajectories which lead to stable
renormalized propagators. This subset is in one-to-one correspondence to the
complete renormalization group trajectories obtained in computations which do
not track of the higher-derivative terms. Thus our asymptotically safe
gravity-matter systems are not haunted by Ostrogradski ghosts.
