  Accurate cosmology from upcoming weak lensing surveys relies on knowledge of
the total matter power spectrum at percent level at scales $k < 10$ $h$/Mpc,
for which modelling the impact of baryonic physics is crucial. We compare
measurements of the total matter power spectrum from the Horizon cosmological
hydrodynamical simulations: a dark matter-only run, one with full baryonic
physics, and another lacking Active Galactic Nuclei (AGN) feedback. Baryons
cause a suppression of power at $k\simeq 10$ $h/$Mpc of $<15\%$ at $z=0$, and
an enhancement of a factor of a few at smaller scales due to the more efficient
cooling and star formation. The results are sensitive to the presence of the
highest mass haloes in the simulation and the distribution of dark matter is
also impacted up to a few percent. The redshift evolution of the effect is
non-monotonic throughout $z=0-5$ due to an interplay between AGN feedback and
gas pressure, and the growth of structure. We investigate the effectiveness of
an analytic `baryonic correction model' in describing our results. We require a
different redshift evolution and propose an alternative fitting function with
$4$ free parameters that reproduces our results within $5\%$. Compared to other
simulations, we find the impact of baryonic processes on the total matter power
spectrum to be smaller at $z=0$. Correspondingly, our results suggest that AGN
feedback is not strong enough in the simulation. Total matter power spectra
from the Horizon simulations are made publicly available at
https://www.horizon-simulation.org/catalogues.html
