The bulk of the X-ray emission in Active Galactic Nuclei (AGN) is produced very close to the accreting supermassive black hole (SMBH), in a corona of hot electrons which up scatters optical and ultraviolet photons from the accretion flow. The cutoff energy ($E_{rm C}$) of the primary X-ray continuum emission carries important information on the physical characteristics of the X-ray emitting plasma, but little is currently known about its potential relation with the properties of accreting SMBHs. Using the largest broad-band (0.3-150 keV) X-ray spectroscopic study available to date, we investigate how the corona is related to the AGN luminosity, black hole mass and Eddington ratio ($lambda_{rm Edd}$). Assuming a slab corona the median values of the temperature and optical depth of the Comptonizing plasma are $kT_{rm e}=105 pm 18$ keV and $tau=0.25pm0.06$, respectively. When we properly account for the large number of $E_{rm C}$ lower limits, we find a statistically significant dependence of the cutoff energy on the Eddington ratio. In particular, objects with $ lambda_{rm Edd}>0.1$ have a significantly lower median cutoff energy ($E_{rm C}=160pm41$ keV) than those with $lambda_{rm Edd}leq 0.1$ ($E_{rm C}=370pm51$ keV). This is consistent with the idea that radiatively compact coronae are also cooler, because they tend to avoid the region in the temperature-compactness parameter space where runaway pair production would dominate. We show that this behaviour could also straightforwardly explain the suggested positive correlation between the photon index ($Gamma$) and the Eddington ratio, being able to reproduce the observed slope of the $Gamma-lambda_{rm Edd}$ trend.
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