  In this paper we introduce and study the primitive equations with
$\textit{non}$-isothermal turbulent pressure and transport noise. They are
derived from the Navier-Stokes equations by employing stochastic versions of
the Boussinesq and the hydrostatic approximations. The temperature dependence
of the turbulent pressure can be seen as a consequence of an additive noise
acting on the small vertical dynamics. For such model we prove global
well-posedness in $H^1$ where the noise is considered in both the It\^{o} and
Stratonovich formulations. Compared to previous variants of the primitive
equations, the one considered here present a more intricate coupling between
the velocity field and the temperature. The corresponding analysis is seriously
more involved than in the deterministic setting. Finally, the continuous
dependence on the initial data and the energy estimates proven here are new,
even in the case of isothermal turbulent pressure.
