  Cosmological analyses based on surveys of galaxy clusters observed through
the Sunyaev-Zel'dovich (SZ) effect strongly rely on the mean pressure profile
of the cluster population. A tension is currently observed between the
cosmological constraints obtained from the analyses of the CMB primary
anisotropies and those from cluster abundance in SZ surveys. This discrepancy
may be explained by a wrong estimate of the hydrostatic bias parameter that
links the hydrostatic mass to the true mass of galaxy clusters. However, a
variation of both the amplitude and the shape of the mean pressure profile
could also explain part of this tension. We analyze the effects of a
modification of this profile on the constraints of the $\sigma_8$ and
$\Omega_m$ parameters through the analysis of the SZ power spectrum measured by
the $Planck$ collaboration. We choose two mean pressure profiles that are
respectively lower and higher than the one obtained from the observation of
nearby clusters by $Planck$. The selection of the parameters of these two
profiles is based on the current estimates of the pressure and gas mass
fraction profile distributions at low redshift. The cosmological parameters
found for these two profiles are significantly different from the ones obtained
with the $Planck$ pressure profile. We conclude that a ${\sim}15\%$ decrease of
the amplitude of the mean normalized pressure profile would alleviate the
tension observed between the constraints of $\sigma_8$ and $\Omega_m$ from the
CMB and cluster analyses without requiring extreme values of the mass bias
parameter.
