  We derive X-ray mass, luminosity, and temperature profiles for 45 galaxy
clusters to explore relationships between halo mass, AGN feedback, and central
cooling time. We find that radio--mechanical feedback power (referred to here
as "AGN power") in central cluster galaxies correlates with halo mass as
P$_{\rm mech}$ $\propto$ M$^{1.55\pm0.26}$, but only in halos with central
atmospheric cooling times shorter than 1 Gyr. The trend of AGN power with halo
mass is consistent with the scaling expected from a self-regulating AGN
feedback loop, as well as with galaxy and central black hole co-evolution along
the $M_{\rm BH} - \sigma$ relation. AGN power in clusters with central
atmospheric cooling times longer than $\sim 1$ Gyr typically lies two orders of
magnitude below those with shorter central cooling times. Galaxies centred in
clusters with long central cooling times nevertheless experience ongoing and
occasionally powerful AGN outbursts. We further investigate the impact of
feedback on cluster scaling relations. We find $L-T$, and $M-T$ relations in
clusters with direct evidence of feedback which are steeper than self-similar,
but not atypical compared to previous studies of the full cluster population.
While the gas mass rises, the stellar mass remains nearly constant with rising
total mass, consistent with earlier studies. This trend is found regardless of
central cooling time, implying tight regulation of star formation in central
galaxies as their halos grew, and long-term balance between AGN heating and
atmospheric cooling. Our scaling relations are presented in forms that can be
incorporated easily into galaxy evolution models.
