  The magnetothermodynamics of skyrmion type matter described by the gauged BPS
baby Skyrme model at zero temperature is investigated. We prove that the BPS
property of the model is preserved also for boundary conditions corresponding
to an asymptotically constant magnetic field. The BPS bound and the
corresponding BPS equations saturating the bound are found. Further, we show
that one may introduce pressure in the gauged model by a redefinition of the
superpotential. Interestingly, this is related to non-extremal type solutions
in the so-called fake supersymmetry method. Finally, we compute the equation of
state of magnetized BSP baby skyrmions inserted into an external constant
magnetic field $H$ and under external pressure $P$, i.e., $V=V(P,H)$, where $V$
is the "volume" (area) occupied by the skyrmions. We show that the BPS baby
skyrmions form a ferromagnetic medium.
