In this study we are motivated by recent realization that the
deconfined phase is NOT a simple plasma of quarks and gluons
when $T\sim (1-2)T_c$, but rather a more strongly interacting
matter. Early lattice calculations pointed to this direction some time
ago when a large deficit in the pressure and entropy of the pure
SU(3) gauge theory as compared to the free gas value was found even
at $T\simeq 4T_c$. Here we ask whether this deficit is a dynamical
feature not just of SU(3) but of all SU(N) gauge theories -- and in
particular whether it survives the N->infinity limit. We find that
the pressure plots of SU(4), and SU(8) are very close to each
other, and not far from the pressure plot of SU(3). This suggests
that the pressure (and entropy) deficit is present in the planar
theory, which is fortunate since in this limit the theory considerably
simplifies, and what happens there should strongly constrain the
possible dynamics underlying the phenomenon.