  Observers have demonstrated that it is now feasible to measure the cosmic
microwave background (CMB) temperature at high redshifts. We explore the
possible constraints on cosmology which might ultimately be derived from such
measurements. Besides providing a consistency check on standard and alternative
cosmologies, possibilities include: constraints on the inhomogeneity and
anisotropy of the universe at intermediate redshift $z ^<_\sim 10$; an
independent probe of peculiar motions with respect to the Hubble flow; and
constraining the epoch of reionization. We argue that the best possibility is
as a probe of peculiar motions. We show, however, that the current measurement
uncertainty ($\Delta T= \pm 0.002 $K) in the local present absolute CMB
temperature imposes intrinsic limits on the use of such CMB temperature
measurements as a cosmological probe. At best, anisotropies at intermediate
redshift could only be constrained at a level of $^>_\sim 0.1%$ and peculiar
motions could only be determined to an uncertainty of $^>_\sim 311$ km
s$^{-1}$. If the high $z$ CMB temperature can only be measured with a precision
comparable to the uncertainty of the local interstellar CMB temperature, then
peculiar motions could be determined to an uncertainty of $1101 (1+z)^{-1}
[\Delta T_{CMB}(z)/0.01 K] km s^{-1}$.
