  Horndeski gravity is a popular contender for a phenomenological model of
dynamical dark energy, and as such subject to observational constraints. In
this work, we ask whether Horndeski gravity can be more than a phenomenological
model and instead become a fundamental theory, which extends towards high
energy scales and includes quantum effects. We find that within the
asymptotic-safety paradigm, an ultraviolet completion of a simple class of
models of Horndeski gravity is achievable, but places strong constraints on the
couplings of the theory. These constraints are not compatible with dynamical
dark energy. Further, we find a similar result in an effective-field theory
approach to this class of models of Horndeski gravity: under the assumption
that there is no new strongly-coupled physics below the Planck scale, quantum
gravity fluctuations force the Horndeski couplings to be too small to achieve
an explanation of dynamical dark energy.
