The cosmic microwave background (CMB) monopole temperature evolves with the inverse of the cosmological scale factor, independent of many cosmological assumptions. With sufficient sensitivity, real-time cosmological observations could thus be used to measure the local expansion rate of the Universe using the cooling of the CMB. We forecast how well a CMB spectrometer could determine the Hubble constant via this method. The primary challenge of such a mission lies in the separation of Galactic and extra-Galactic foreground signals from the CMB at extremely high precision. However, overcoming these obstacles could potentially provide an independent, highly robust method to shed light on the current low-/high-$z$ Hubble tension. An experiment with 3000 linearly spaced bins between 5 GHz and 3 THz with a sensitivity of 1 $\mathrm{mJy\sqrt{yr}~sr^{-1}}$ per bin, could measure $H_0$ to 3% over a 10 year mission, given current foreground complexity. This sensitivity would also enable high-precision measurements of the expected $\Lambda$CDM spectral distortions, but remains futuristic at this stage.
Comment: Authors' accepted version uploaded. (Published in ApJ)