  The redshift dependence of the cosmic microwave background temperature,
$T(z)=T_0(1+z)$, is a key prediction of standard cosmology, but this relation
is violated in many extensions thereof. Current astrophysical facilities can
probe it in the redshift range $0\le z\le6.34$. We extend recent work by Gelo
{\it et al.} (2022) showing that for several classes of models (all of which
aim to provide alternative mechanisms for the recent acceleration of the
universe) the constraining power of these measurements is comparable to that of
other background cosmology probes. Specifically, we discuss constraints on two
classes of models not considered in the earlier work: a model with torsion and
a recently proposed phenomenological dynamical dark energy model which can be
thought of as a varying speed of light model. Moreover, for both these models
and also for those in the earlier work, we discuss how current constraints may
be improved by next-generation ground and space astrophysical facilities.
Overall, we conclude that these measurements have a significant cosmological
impact, mainly because they often constrain combinations of model parameters
that are orthogonal, in the relevant parameter space, to those of other probes.
