||Accurate cosmology from upcoming weak lensing surveys relies on knowledge of the total matter power spectrum at per cent level at scales k < 10 hMpc(-1), 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-matteronly run, one with full baryonic physics, and another lacking active galactic nucleus (AGN) feedback. Baryons cause a suppression of power at k similar or equal to 10 h Mpc(-1) of <15 per cent 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 per cent. 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 four free parameters that reproduces our results within 5 per cent. 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.