  Radio-mode feedback associated with the active galactic nuclei (AGN) at the
cores of galaxy clusters injects large amount of energy into the intracluster
medium (ICM), offsetting radiative losses through X-ray emission. This
mechanism prevents the ICM from rapidly cooling down and fueling extreme
starburst activity as it accretes onto the central galaxies, and is therefore a
key ingredient in the evolution of galaxy clusters. However, the influence and
mode of feedback at high redshifts (z~1) remains largely unknown. Low-frequency
sub-arcsecond resolution radio observations taken with the International LOFAR
Telescope have demonstrated their ability to assist X-ray observations with
constraining the energy output from the AGNs (or "cavity power") in galaxy
clusters, thereby enabling research at higher redshifts than before. In this
pilot project, we test this hybrid method on a high redshift (0.6<z<1.3) sample
of 13 galaxy clusters for the first time with the aim of verifying the
performance of this method at these redshifts and providing the first estimates
of the cavity power associated with the central AGN for a sample of distant
clusters. We were able to detect clear radio lobes in three out of thirteen
galaxy clusters at redshifts 0.7<z<0.9, and use these detections in combination
with ICM pressures surrounding the radio lobes obtained from standard profiles
to calculate the corresponding cavity powers of the AGNs. By combining our
results with the literature, the current data appear to suggest that the
average cavity power peaked at a redshift of z~0.4 and slowly decreases toward
higher redshifts. However, we require more and tighter constraints on the
cavity volume and a better understanding of our observational systematics to
confirm any deviation of the cavity power trend from a constant level.
