Nuclear hard X-ray luminosities (L X,nuc ) for a sample of 112 early type galaxies within a distance of 67 Mpc are used to investigate their relationship with the central galactic black hole mass M BH (coming from direct dynamical studies or the M BH −σ relation), the inner galactic structure (using the parameters describing its cuspiness), the hot gas content and the core radio luminosity. For this sample, L X,nuc ranges from 10 38 to 10 42 erg s −1 , and the Eddington ratio L X,nuc /L Edd from 10 −9 to 10 −4 , with the largest values belonging to four Seyfert galaxies. Together with a trend for L X,nuc to increase on average with the galactic luminosity L B and M BH , there is a wide variation of L X,nuc (and L X,nuc /L Edd ), by up to 4 orders of magnitude, at any fixed L B > 6 × 10 9 L B,⊙ or M BH > 10 7 M ⊙ . This large observed range should reflect a large variation of the mass accretion rateṀ BH , and possible reasons for this difference are searched for. On the circumnuclear scale, in a scenario where accretion is (quasi) steady,Ṁ BH at fixed L B (or M BH ) could vary due to differences in the fuel production rate from stellar mass return linked to the inner galactic structure; a trend of L X,nuc with cuspiness is not observed, though, while a tendency for L X,nuc /L Edd to be larger in cuspier galaxies is present. In fact,Ṁ BH is predicted to vary with cuspiness by a factor exceeding a few only in hot gas poor galaxies and for large differences in the core radius; for a subsample with these characteristics the expected effect seems to be present in the observed L X,nuc values. L X,nuc does not show a dependence on the age of the stellar population in the central galactic region, for ages>3 Gyr; less luminous nuclei, though, are found among the youngest galaxies or galaxies with a younger stellar component. On the global galactic scale, L X,nuc shows a trend with the total galactic hot gas cooling rate (L X,ISM ): it is detected both in gas poor and gas rich galaxies, and on average increases with L X,ISM , but again with a large scatter. The observed lack of a tight relationship between L X,nuc and the circumnuclear and total gas content can be explained if accretion is regulated by factors overcoming the importance of fuel availability, as 1) the gas is heated by black hole feedback andṀ BH varies due to an activity cycle, and 2) the mass effectively accreted by the black hole can be largely reduced with respect to that entering the circumnuclear region, as in radiatively inefficient accretion with winds/outflows. Finally, differently from L X,nuc , the central 5 GHz VLA luminosity shows a clear trend with the inner galactic structure, that is similar to that shown by the total soft X-ray emission; therefore it is suggested that they could both be produced by the hot gas.