  The cosmic microwave background (CMB) energy spectrum is a near-perfect
blackbody. The standard model of cosmology predicts small spectral distortions
to this form, but no such distortion of the sky-averaged CMB spectrum has yet
been measured. We calculate the largest expected distortion, which arises from
the inverse Compton scattering of CMB photons off hot, free electrons, known as
the thermal Sunyaev-Zel'dovich (tSZ) effect. We show that the predicted signal
is roughly one order of magnitude below the current bound from the COBE-FIRAS
experiment, but can be detected at enormous significance ($\gtrsim 1000\sigma$)
by the proposed Primordial Inflation Explorer (PIXIE). Although cosmic variance
reduces the effective signal-to-noise to $230\sigma$, this measurement will
still yield a sub-percent constraint on the total thermal energy of electrons
in the observable universe. Furthermore, we show that PIXIE can detect subtle
relativistic effects in the sky-averaged tSZ signal at $30\sigma$, which
directly probe moments of the optical depth-weighted intracluster medium
electron temperature distribution. These effects break the degeneracy between
the electron density and temperature in the mean tSZ signal, allowing a direct
inference of the mean baryon density at low redshift. Future spectral
distortion probes will thus determine the global thermodynamic properties of
ionized gas in the universe with unprecedented precision. These measurements
will impose a fundamental "integral constraint" on models of galaxy formation
and the injection of feedback energy over cosmic time.
