We present a detailed diagnostic study of the observed temperatures of the hot X-ray coronae of early-type galaxies. By extending the investigation carried out in Pellegrini (2011) with spherical models, we focus on the dependence of the energy budget and temperature of the hot gas on the galaxy structure and internal stellar kinematics. By solving the Jeans equations we construct realistic axisymmetric three-component galaxy models (stars, dark matter halo, central black hole) with different degrees of flattening and rotational support. The kinematical fields are projected along different lines of sight, and the aperture velocity dispersion is computed within a fraction of the circularized effective radius. The model parameters are chosen so that the models resemble real ETGs and lie on the Faber-Jackson and Size-Luminosity relations. For these models we compute T_* (the stellar heating contribution to the gas injection temperature) and T_gm (the temperature equivalent of the energy required for the gas escape). In particular, different degrees of thermalisation of the ordered rotational field of the galaxy are considered. We find that T_* and T_gm can vary only mildly due to a pure change of shape. Galaxy rotation instead, when not thermalised, can lead to a large decrease of T_*; this effect can be larger in flatter galaxies that can be more rotationally supported. Recent temperature measurements T_x, obtained with Chandra, are larger than, but close to, the T_* values of the models, and show a possible trend for a lower T_x in flatter and more rotationally supported galaxies; this trend can be explained by the lack of thermalisation of the whole stellar kinetic energy. Flat and rotating galaxies also show lower L_x values, and then a lower gas content, but this is unlikely to be due to the small variation of T_gm found here for them.
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