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The statistics of thermal gas pressure are a new and promising probe of cosmology and astrophysics. The large-scale cross-correlation between galaxies and the thermal Sunyaev-Zeldovich effect gives the bias-weighted mean electron pressure, $\langle b_\mathrm{h}P_e\rangle$. In this paper, we show that $\langle b_\mathrm{h}P_e\rangle$ is sensitive to the amplitude of fluctuations in matter density, for example $\langle b_\mathrm{h}P_e\rangle\propto \left(\sigma_8\Omega_\mathrm{m}^{0.81}h^{0.67}\right)^{3.15}$ at redshift $z=0$.
We find that at $z<0.5$ the observed $\langle b_\mathrm{h}P_e\rangle$ is smaller than that predicted by the state-of-the-art hydrodynamical simulations of galaxy formation, MillenniumTNG, by a factor of $0.93$.
%We find that state-of-the-art hydrodynamical simulations of galaxy formation, \emph{Magneticum} and MillenniumTNG, predict a larger $\langle b_\mathrm{h}P_e\rangle$ than that observed at $z<0.5$ by a factor of $1.05$.
This can be explained by a lower value of $\sigma_8$ and $\Omega_\mathrm{m}$, similar to the so-called ``$S_8$ tension'' seen in the gravitational lensing effect. The difference between \emph{Magneticum} and MillenniumTNG at $z<2$ is small, indicating that the difference in the galaxy formation models used by these simulations has little impact on $\langle b_\mathrm{h}P_e\rangle$ at this redshift range. At higher $z$, we find that both simulations are in a modest tension with the existing upper bounds on $\langle b_\mathrm{h}P_e\rangle$. We also find a significant difference between these simulations, which we attribute to the difference in the galaxy formation models. Therefore, more precise measurements of $\langle b_\mathrm{h}P_e\rangle$ at $z>2$ will provide a new test of our understanding of galaxy formation, and those at $z<2$ will be a powerful probe of cosmology.