  Measurements of the thermal Sunyaev-Zel'dovich (tSZ) effect have long been
recognized as a powerful cosmological probe. Here we assess the importance of
relativistic temperature corrections to the tSZ signal on the power spectrum
analysis of the Planck Compton-$y$ map, developing a novel formalism to account
for the associated effects. The amplitude of the tSZ power spectrum is found to
be sensitive to the effective electron temperature, $\bar{T}_e$, of the cluster
sample. Omitting the corresponding modifications leads to an underestimation of
the $yy$-power spectrum amplitude. Relativistic corrections thus add to the
error budget of tSZ power spectrum observables such as $\sigma_8$. This could
help alleviate the tension between various cosmological probes, with the
correction scaling as $\Delta \sigma_8/\sigma_8 \simeq
0.019\,[k\bar{T}_e\,/\,5\,{\rm keV}]$ for Planck. At the current level of
precision, this implies a systematic shift by $\simeq 1\sigma$, which can also
be interpreted as an overestimation of the hydrostatic mass bias by $\Delta b
\simeq 0.046\,(1-b)\,[k\bar{T}_e\,/\,5\,{\rm keV}]$, bringing it into better
agreement with hydrodynamical simulations. It is thus time to consider
relativistic temperature corrections in the processing of current and future
tSZ data.
