  The measurement of present-day temperature of the Cosmic Microwave Background
(CMB), $T_0 = 2.72548 \pm 0.00057$ K (1$\sigma$), made by the Far-InfraRed
Absolute Spectrophotometer (FIRAS), is one of the most precise measurements
ever made in Cosmology. On the other hand, estimates of the Hubble Constant,
$H_0$, obtained from measurements of the CMB temperature fluctuations assuming
the standard $\Lambda$CDM model exhibit a large ($4.1\sigma$) tension when
compared with low-redshift, model-independent observations. Recently, some
authors argued that a slightly change in $T_0$ could alleviate or solve the
$H_0$-tension problem. Here, we investigate evidence for a hotter or colder
universe by performing an independent analysis from currently available
temperature-redshift $T(z)$ measurements. Our analysis (parametric and
non-parametric) shows a good agreement with the FIRAS measurement and a
discrepancy of $\gtrsim 1.9\sigma$ from the $T_0$ values required to solve the
$H_0$ tension. This result reinforces the idea that a solution of the
$H_0$-tension problem in fact requires either a better understanding of the
systematic errors on the $H_0$ measurements or new physics.
