The analysis of a deep (579ks) Chandra ACIS pointing of the elliptical galaxy NGC4278, which hosts a low-luminosity active galactic nucleus (AGN) and compact radio emission, allowed us to detect extended emission from hot gas out to a radius of 5kpc, with 0.5-8keV luminosity of 2.4 × 10 erg/s. The emission is elongated in the NE-SW direction, misaligned with respect to the stellar body, and aligned with the ionized gas and with the Spitzer IRAC 8 μm non-stellar emission. The nuclear X-ray luminosity decreased by a factor of 18 since the first Chandra observation in 2005, a dimming that enabled the detection of hot gas even at the position of the nucleus. The gas shows a significantly larger temperature (kT = 0.75keV) in both the projected and deprojected profiles in the inner 300pc than in the surrounding region, where it stays at 0.3keV, a value lower than expected from standard gas heating assumptions. The nuclear X-ray emission is consistent with that of a low radiative efficiency accretion flow, accreting mass at a rate close to the Bondi rate; estimates of the power of the nuclear jets require that the accretion rate is not largely reduced with respect to the Bondi rate. Among possible origins for the central large hot gas temperature, such as gravitational heating from the central massive black hole and a recent AGN outburst, interaction with the nuclear jets seems more likely, especially if the latter remain confined, and heat the nuclear region frequently. The unusual hot gas distribution on the galactic scale could be due to the accreting cold gas triggering the cooling of the hot phase, a process also contributing to the observed line emission from ionized gas, and to the hot gas temperature being lower than expected; alternatively, the latter could be due to the efficiency of the Type Ia supernova heating that is lower than usually adopted.