Recently, the temperature T and luminosity L_X of the hot gas halos of early type galaxies have been derived with unprecedented accuracy from Chandra data, for 30 galaxies covering a wider range of galactic luminosity (and central velocity dispersion sigma_c) than before. This work investigates the origin of the observed temperatures, by examining the relationship between them and the galaxy structure, the gas heating due to Type Ia supernovae (SNIas) and the gravitational potential, and the dynamical status of the gas flow. In galaxies with sigma_c<200 km/s, the Ts are close to a fiducial average temperature for the gas when in outflow; at 200<sigma_c (km/s)<250, the Ts are generally lower than this, and unrelated with sigma_c, which requires a more complex gas flow status; at larger sigma_c, the Ts may increase as sigma_c^2, as expected for infall heating, though heating from SNIas, independent of sigma_c, should be dominant. All observed Ts are larger than the virial temperature, by up to ~0.5 keV. This additional heating can be provided in the X-ray brightest galaxies by SNIas and infall heating, with a SNIas energy input even lower than in standard assumptions; in the X-ray fainter ones it can be provided by SNIas, whose energy input would be required close to the full standard value at the largest sigma_c. This same energy input, though, would produce temperatures larger than observed at low sigma_c, if entirely thermalized. The values of the observed Ts increase from outflows to inflows; the gas is relatively hotter in outflows, though, if the Ts are rescaled by the virial temperature. For 200<sigma_c(km/s)<250, lower L_X values tend to correspond to lower Ts, which deserves further investigation.
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