  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.
