We present detailed spatially resolved measurements of the thermodynamic properties of the X-ray-emitting gas in the inner regions of the five nearest, X-ray and optically brightest, and most X-ray morphologically relaxed giant elliptical galaxies known. Beyond the innermost region at r ≳ 1 kpc, and out to r ∼ 6 kpc, the density, pressure, entropy and cooling time distributions for the X-ray-emitting gas follow remarkably similar, simple, power-law-like distributions. Notably, the entropy profiles follow a form K ∝ rα, with an index α = 0.92–1.07. The cumulative hot X-ray-emitting gas mass profiles and the gas mass to stellar light ratios of all five galaxies are also similar. Overall the observed similarity of the thermodynamic profiles in this radial range argues that, in these systems, relativistic jets heat the gas at a similar rate averaged over time-scales longer than the cooling time tcool ≳ 108 yr. These jets are powered by accretion from the hot gas, or material entrained within it, on to the central supermassive black hole. This jet heating creates an energy balance where heating and cooling are in equilibrium, keeping the hot galactic atmospheres in a ‘steady state’. Within r ≲ 1 kpc, this similarity breaks down: the observed entropy profiles show well-resolved flattening and the values differ from system to system substantially. The accretion rate on to the black hole and the active galactic nucleus activity, heating the interstellar medium, must therefore vary significantly on time-scales shorter than tcool = 107–108 yr.