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 ($4.1sigma$) 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 $gtrsim 1.9sigma$ 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.
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