  The cosmic microwave background temperature is a cornerstone astrophysical
observable. Its present value is tightly constrained, but its redshift
dependence, which can now be determined until redshift $z\sim6.34$, is also an
important probe of fundamental cosmology. We show that its constraining power
is now comparable to that of other background cosmology probes, including Type
Ia supernovae and Hubble parameter measurements. We illustrate this with three
models, each based on a different conceptual paradigm, which aim to explain the
recent acceleration of the universe. We find that for parametric extension of
$\Lambda$CDM the combination of temperature and cosmological data significantly
improves constraints on the model parameters, while for alternative models
without a $\Lambda$CDM limit this data combination rules them out.
