temperature of a molecule in a galaxy

Oh yeah, good point malawi, I think my mind simply converted 'molecule' to 'gas' when I read the OP, since as you say, a single molecule can't have a temperature!

VLT Observations Confirm that the Universe Was Hotter in the Past
UVES Measures the Cosmic Temperature 12 Billion Years Ago

Summary

A fundamental prediction of the Big Bang theory has finally been verified .

For the first time, an actual measurement has been made of the temperature of the cosmic microwave background radiation, at a time when the Universe was only about 2.5 billion years old . This fundamental and very difficult observation was achieved by a team of astronomers from India, France and ESO [1]. They obtained a detailed spectrum of a quasar in the distant Universe, using the UV-Visual Echelle Spectrograph (UVES) instrument at the ESO 8.2-m VLT KUEYEN telescope at the Paranal Observatory.

If the Universe was indeed formed in a Big Bang, as most astrophysicists believe, the glow of this primeval fireball should have been warmer in the past. This is exactly what is found by the new measurements.

The analysis of the VLT spectrum of the distant quasar not only gives the definitive proof of the presence of the relict radiation in the early Universe, it also shows that it was indeed significantly warmer than it is today, as predicted by the theory.

PR Photo 35/00 : VLT spectrum of the distant quasar PKS 1232+0815 , displaying lines of carbon atoms from an intervening cloud in which the cosmic temperature was measured.

The microwave background temperature at the redshift of 2.33771
Authors: R. Srianand (IUCAA, Pune), Patrick Petitjean (IAP, Paris), Cedric Ledoux (ESO, Munich)
(Submitted on 11 Dec 2000)

Abstract: The Cosmic Microwave Background radiation is a fundamental prediction of Hot Big Bang cosmology. The temperature of its black-body spectrum has been measured at the present time, $T_{\rm CMBR,0}$ = 2.726$\pm$ 0.010 K, and is predicted to have been higher in the past. At earlier time, the temperature can be measured, in principle, using the excitation of atomic fine structure levels by the radiation field. All previous measurements however give only upper limits as they assume that no other significant source of excitation is present. Here we report the detection of absorption from the first {\sl and} second fine-structure levels of neutral carbon atoms in an isolated remote cloud at a redshift of 2.33771. In addition, the unusual detection of molecular hydrogen in several rotational levels and the presence of ionized carbon in its excited fine structure level make the absorption system unique to constrain, directly from observation, the different excitation processes at play. It is shown for the first time that the cosmic radiation was warmer in the past. We find 6.0 < T_{\rm CMBR} < 14 K at z = 2.33771 when 9.1 K is expected in the Hot Big Bang cosmology.