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$^{1.55pm0.26}$, 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.
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