  The linear increase of the cosmic microwave background (CMB) temperature with
cosmological redshift, $T_{\rm CMB} = T_0(1 + z)$, is a prediction of the
standard cosmological $\Lambda$CDM model. There are currently two methods to
measure this dependence at redshift $z>0$, and that is equally important to
estimate the CMB temperature $T_0$ at the present epoch $z=0$. The first method
is based on the Sunyaev-Zeldovich (SZ) effect for a galaxy cluster. aThe second
method is based on the analysis of the populations of atomic and molecular
energy levels observed in the absorption spectra of quasars. This method allows
$T_{\rm CMB}(z)$ to be measured directly. We present new estimates of $T_{\rm
CMB}(z_i)$ in the redshift range $1.7\le z_i \le3.3$ based on the analysis of
excitation of the CO rotational levels and C\,{\sc i} fine-structure levels in
15 absorption systems. We take into account collisional excitation of CO and
C\,{\sc i} with hydrogen atoms and H$_2$ and radiative pumping of C\,{\sc i} by
the interstellar ultraviolet radiation. Applying this corrections leads to a
systematic decrease in the previously obtained estimates of $T_{\rm CMB}(z_i)$
(for some systems the magnitude of the effect is $\sim$10\%). Combining our
measurements with the measurements of $T_{\rm CMB}(z)$ in galaxy clusters we
have obtained a constraint on the parameter $\beta=+0.010\pm0.013$, which
characterizes the deviation of the CMB temperature from the standard relation,
$T_{\rm CMB} = T_0(1 + z)^{1-\beta}$, and an independent estimate of the CMB
temperature at the present epoch, $T_0 = 2.719\pm0.009$\,K, which agrees well
with the estimate from orbital measurements, $T_0 = 2.7255\pm0.0006$\,K. This
independent estimate is very important because it was obtained using
cosmological data, in contrast to satellite measurements, which are obtained
"here" and "now".
