  With increasing sensitivity, angular resolution, and frequency coverage,
future cosmic microwave background (CMB) experiments like PICO will allow us to
access new information about galaxy clusters through the relativistic thermal
Sunyaev-Zeldovich (SZ) effect. We will be able to map the temperature of
relativistic electrons across the entire sky, going well beyond a simple
detection of the relativistic SZ effect by cluster stacking methods that
currently define the state-of-the-art. Here, we propose a new map-based
approach utilizing SZ-temperature moment expansion and constrained-ILC methods
to extract electron gas temperature maps from foreground-obscured CMB data.
This delivers a new independent map-based observable, the electron temperature
power spectrum $T_{\rm e}^{yy}(\ell)$, which can be used to constrain cosmology
in addition to the Compton-$y$ power spectrum $C_\ell^{yy}(\ell)$ . We find
that PICO has the required sensitivity, resolution, and frequency coverage to
accurately map the electron gas temperature of galaxy clusters across the full
sky, covering a broad range of angular scales. Frequency-coverage at
$\nu\gtrsim 300\,{\rm GHz}$ plays an important role for extracting the
relativistic SZ effect in the presence of foregrounds. For Coma, PICO will
allow us to directly reconstruct the electron temperature profile using the
relativistic SZ effect. Coma's average electron temperature will be measured to
$10\sigma$ significance after foreground removal using PICO. Low-angular
resolution CMB experiment like LiteBIRD could achieve $2\sigma$ to $3\sigma$
measurement of the electron temperature of this largest cluster. Our analysis
highlights a new spectroscopic window into the thermodynamic properties of
galaxy clusters and the diffuse electron gas at large angular scales.
