  We present a detailed study of the thermodynamic properties of the
intracluster medium of the only low redshift galaxy cluster to contain a highly
luminous quasar, H1821+643. The cluster is a highly massive, strong cool core
cluster. We find that the ICM entropy around the quasar is significantly lower
than that of other similarly massive strong cool core clusters within the
central 80 kpc, and that the entropy lies significantly below the extrapolated
baseline entropy profile from hierarchical structure formation. By comparing
the scaled temperature profile with those of other strong cool core clusters of
similar total mass, we see that the entropy deficiency is due to the central
temperature being significantly lower. This suggests that the presence of the
quasar in the core of H1821+643 has had a dramatic cooling effect on the
intracluster medium around it. We find that, if the quasar was brighter in the
past, Compton cooling by radiation from the quasar may have caused the low
entropy and temperature levels in the ICM around the quasar. Curiously, the
gradients of the steep central temperature and entropy decline are in
reasonable agreement with the profiles expected for a constant pressure cooling
flow. It is possible that the system has been locked into a Compton cooled
feedback cycle which prevents energy release from the black hole heating the
gas sufficiently to switch it off, leading to the formation of a huge (~3x10^10
solar mass) supermassive black hole.
