  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.
