We study the observed relation between accretion rate (in terms of L/L_Edd) and shape of the hard X-ray spectral energy distribution (namely the photon index Gamma_X) for a large sample of 228 hard X-ray selected, low-redshift active galactic nuclei (AGN), drawn from the Swift/BAT AGN Spectroscopic Survey (BASS). This includes 30 AGN for which black hole mass (and therefore L/L_Edd) is measured directly through masers, spatially resolved gas or stellar dynamics, or reverberation mapping. The high quality and broad energy coverage of the data provided through BASS allow us to examine several alternative determinations of both Gamma_X and L/L_Edd. For the BASS sample as a whole, we find a statistically significant, albeit very weak correlation between Gamma_X and L/L_Edd. The best-fitting relations we find, Gamma_X=0.15 log(L/L_Edd)+const., are considerably shallower than those reported in previous studies. Moreover, we find no corresponding correlations among the subsets of AGN with different M_BH determination methodology. In particular, we find no robust evidence for a correlation when considering only those AGN with direct or single-epoch M_BH estimates. This latter finding is in contrast to several previous studies which focused on z>0.5 broad-line AGN. We discuss this tension and conclude that it can be partially accounted for if one adopts a simplified, power-law X-ray spectral model, combined with L/L_Edd estimates that are based on the continuum emission and on single-epoch broad line spectroscopy in the optical regime. We finally highlight the limitations on using Gamma_X as a probe of supermassive black hole evolution in deep extragalactic X-ray surveys.
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