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, T¯e, of the cluster sample. Omitting the corresponding modifications leads to an underestimation of the yypower spectrum amplitude. Relativistic corrections thus add to the error budget of tSZ power spectrum observables such as σ8. This could help alleviate the tension between various cosmological probes, with the correction scaling as Δσ8/σ8≃0.019[kT¯e/5keV] for Planck. At the current level of precision, this implies a systematic shift by ≃ 1σ, which can also be interpreted as an overestimation of the hydrostatic mass bias by Δb≃0.046(1−b)[kT¯e/5keV], 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.