  We discuss the possibility to constrain the relation between redshift and
temperature of the cosmic microwave background (CMB) using multifrequency
Sunyaev-Zeldovich (SZ) observations. We have simulated a catalog of clusters of
galaxies detected through their SZ signature assuming the sensitivities that
will be achieved by the {\it Planck} satellite at 100, 143 and 353 GHz, taking
into account the instrumental noise and the contamination from the Cosmic
Infrared Background and from unresolved radiosources. We have parametrized the
cosmological temperature-redshift law as $T\propto (1+z)^{(1-a)}$. Using two
sets of SZ flux density ratios (100/143 GHz, which is most sensitive to the
parametrization of the $T-z$ law, and 143/353 GHz, which is most sensitive to
the peculiar velocities of the clusters) we show that it is possible to recover
the $T-z$ law assuming that the temperatures and redshifts of the clusters are
known. From a simulated catalog of $\sim 1200$ clusters, the parameter $a$ can
be recovered to an accuracy of 10$^{-2}$. Sensitive SZ observations thus appear
as a potentially useful tool to test the standard law. Most cosmological models
predict a linear variation of the CMB temperature with redshift. The discovery
of an alternative law would have profound implications on the cosmological
model, implying creation of energy in a manner that would still maintain the
black-body shape of the CMB spectrum at redshift zero.
