  Compressibility effects in a turbulent transport of temperature field are
investigated applying the quasi-linear approach for small P\'eclet numbers and
the spectral $\tau$ approach for large P\'eclet numbers. Compressibility of a
fluid flow reduces the turbulent diffusivity of the mean temperature field
similarly to that for particle number density and magnetic field. However,
expressions for the turbulent diffusion coefficient for the mean temperature
field in a compressible turbulence are different from those for the mean
particle number density and the mean magnetic field. Combined effect of
compressibility and inhomogeneity of turbulence causes an increase of the mean
temperature in the regions with more intense velocity fluctuations due to a
turbulent pumping. Formally, this effect is similar to a phenomenon of
compressible turbophoresis found previously [J. Plasma Phys. {\bf 84},
735840502 (2018)] for non-inertial particles or gaseous admixtures. Gradient of
the mean fluid pressure results in an additional turbulent pumping of the mean
temperature field. The latter effect is similar to turbulent barodiffusion of
particles and gaseous admixtures. Compressibility of a fluid flow also causes a
turbulent cooling of the surrounding fluid due to an additional sink term in
the equation for the mean temperature field. There is no analog of this effect
for particles.
