  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 $\langle b_\mathrm{h}P_e\rangle\propto
\left(\sigma_8\Omega_\mathrm{m}^{0.81}h^{0.67}\right)^{3.14}$ 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 $0.93$. 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,
although the influence of astrophysics cannot be completely excluded. The
difference between 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 there, which we attribute to a
larger sensitivity to the galaxy formation models in the high redshift regime.
Therefore, more precise measurements of $\langle b_\mathrm{h}P_e\rangle$ at all
redshifts will provide a new test of our understanding of cosmology and galaxy
formation.
