  In the expanding Universe, the average temperature of the cosmic microwave
background (CMB) is expected to depend like TCMB~(1+z) on redshift z. Adiabatic
photon production (or destruction) or deviations from isotropy and homogeneity
could modify this scaling and several observational tests have been carried out
in response. Here, we explain why `adiabatic' conditions are extremely
difficult to establish in the redshift range targeted by these tests. Thus,
instead of leading to a simple rescaling of the CMB temperature, a spectral
distortion should be produced, which can be constrained using COBE/FIRAS. For
scenarios with late photon production, tests of the temperature-redshift
relation (TRR) should therefore be reinterpreted as weak spectral distortion
limits, directly probing the energy dependence of the photon production
process. For inhomogeneous cosmologies, an average y-type distortion is
produced, but this type of distortion can be created in several other ways.
Here, we briefly discuss possible effects that may help disentangling different
contributions to the distortion signal, finding this to be very challenging. We
furthermore argue that tests of the TRR, using the SZ effect have limited
applicability and that for non-gravitational changes to the TRR the CMB
anisotropy spectrum should exhibit an additional y-type dependence.
