  The mean pressure profile of the cluster population is a key element in
cosmological analyses based on surveys of galaxy clusters observed through the
Sunyaev-Zel'dovich (SZ) effect. A variation of both the shape and the amplitude
of this profile could explain part of the discrepancy currently observed
between the cosmological constraints obtained from the analyses of the CMB
primary anisotropies and those from cluster abundance in SZ surveys for a fixed
mass bias parameter. We study the cosmological implications of a modification
of the mean pressure profile through the analysis of the SZ power spectrum
measured by $Planck$. We define two mean pressure profiles on either side of
the one obtained from the observation of nearby clusters by $Planck$. The
parameters of these profiles are chosen to ensure their compatibility with the
distributions of pressure and gas mass fraction profiles observed at low
redshift. We find significant differences between the cosmological parameters
obtained by using these two profiles to fit the $Planck$ SZ power spectrum and
those found in previous analyses. We conclude that a ${\sim}15\%$ decrease of
the amplitude of the mean normalized pressure profile is sufficient to
alleviate the discrepancy observed between the constraints of $\sigma_8$ and
$\Omega_m$ from the CMB and cluster analyses.
