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We derive X-ray mass, luminosity, and temperature profiles for 45 galaxyclusters to explore relationships between halo mass, AGN feedback, and centralcooling time. We find that radio--mechanical feedback power (referred to hereas "AGN power") in central cluster galaxies correlates with halo mass asP$_{\rm mech}$ $\propto$ M$^{1.55\pm0.26}$, but only in halos with centralatmospheric cooling times shorter than 1 Gyr. The trend of AGN power with halomass is consistent with the scaling expected from a self-regulating AGNfeedback loop, as well as with galaxy and central black hole co-evolution alongthe $M_{\rm BH} - \sigma$ relation. AGN power in clusters with centralatmospheric cooling times longer than $\sim 1$ Gyr typically lies two orders ofmagnitude below those with shorter central cooling times. Galaxies centred inclusters with long central cooling times nevertheless experience ongoing andoccasionally powerful AGN outbursts. We further investigate the impact offeedback on cluster scaling relations. We find $L-T$, and $M-T$ relations inclusters 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 risingtotal mass, consistent with earlier studies. This trend is found regardless ofcentral cooling time, implying tight regulation of star formation in centralgalaxies as their halos grew, and long-term balance between AGN heating andatmospheric cooling. Our scaling relations are presented in forms that can beincorporated easily into galaxy evolution models.

A Relationship Between Halo Mass, Cooling, AGN Heating, and the Coevolution of Massive Black Holes