  We investigate a numerical model for AGN feedback where for the first time a
relativistic particle population in AGN-inflated bubbles is followed within a
full cosmological context. In our high-resolution simulations of galaxy cluster
formation, we assume that BH accretion is accompanied by energy feedback that
occurs in two different modes, depending on the accretion rate itself. Unlike
in previous work, we inject a non-thermal particle population of relativistic
protons into the AGN bubbles, instead of adopting a purely thermal heating. We
then follow the subsequent evolution of the cosmic ray (CR) plasma inside the
bubbles, considering both its hydrodynamical interactions and dissipation
processes relevant for the CR population. Due to the different buoyancy of
relativistic plasma and the comparatively long CR dissipation timescale we find
substantial changes in the evolution of clusters as a result of CR feedback. In
particular, the non-thermal population can provide significant pressure support
in central cluster regions at low thermal temperatures, providing a natural
explanation for the decreasing temperature profiles found in cool core
clusters. At the same time, the morphologies of the bubbles and of the induced
X-ray cavities show a striking similarity to observational findings. AGN
feedback with CRs also proves efficient in regulating cluster cooling flows so
that the total baryon fraction in stars becomes limited to realistic values of
the order of 10%. We find that the partial CR support of the intracluster gas
also affects the expected signal of the thermal Sunyaev-Zel'dovich effect, with
typical modifications of the integrated Compton-y parameter within the virial
radius of the order of 10%. [Abridged]
