  The temperature profile of hot gas in galaxies and galaxy clusters is largely
determined by the depth of the total gravitational potential and thereby by the
dark matter (DM) distribution. We use high-resolution hydrodynamical
simulations of galaxy formation to derive a surprisingly simple relation
between the gas temperature and DM properties. We show that this relation holds
not just for galaxy clusters but also for equilibrated and relaxed galaxies at
radii beyond the central stellar-dominated region of typically a few kpc. It is
then clarified how a measurement of the temperature and density of the hot gas
component can lead to an indirect measurement of the DM velocity anisotropy in
galaxies. We also study the temperature relation for galaxy clusters in the
presence of self-regulated, recurrent active galactic nuclei (AGN), and
demonstrate that this temperature relation even holds outside the inner region
of 30 kpc in clusters with an active AGN.
