The relationship between the cosmic microwave background radiation temperature and the redshift, i.e., the $T$--$z$ relation, is examined in a phenomenological dissipative model. The model contains two constant terms, as if a nonzero cosmological constant $��$ and a dissipative process are operative in a homogeneous, isotropic, and spatially flat universe. The $T$--$z$ relation is derived from a general radiative temperature law, as appropriate for describing nonequilibrium states in a creation of cold dark matter (CCDM) model. Using this relation, the radiation temperature in the late universe is calculated as a function of a dissipation rate ranging from $\tilde�� =0$, corresponding to a nondissipative $��$CDM model, to $\tilde�� =1$, corresponding to a fully dissipative CCDM model. The $T$--$z$ relation for $\tilde�� =0$ is linear for standard cosmology and is consistent with observations. However, with increasing dissipation rate $\tilde��$, the radiation temperature gradually deviates from a linear law because the effective equation-of-state parameter varies with time. When the background evolution of the universe agrees with a fine-tuned pure $��$CDM model, the $T$--$z$ relation for low $\tilde��$ matches observations, whereas the $T$--$z$ relation for high $\tilde��$ does not. Previous work also found that a weakly dissipative model accords with measurements of a growth rate for clustering related to structure formations. These results imply that low dissipation is likely for the universe. The weakly dissipative model should be further constrained by recent observations.

Final version accepted for publication in PRD. References are updated. [11 pages, 6 figures, and 1 table]