I confirm your result. More precisely, I find that the ratio PV/<N>kT
for a photon gas is pi^4/(90 zeta(3)). Zeta(3) is about 1.202, so
I find to three sig figs PV/<N>kT = .900.
Is it surprising that PV/NkT is a constant for a photon gas? I don't
think so. The pressure exerted by each photon is proportional to
its momentum, which is proportional to its energy, which is proportional
to the temperature. The factor of V comes partly from the relation
between force and pressure, and partly from the frequency with which
any particular photon will collide with the walls. Think of the usual
microscopic model presented for an ordinary gas of molecules in
introductory physics books. For nonrelativistic molecules it's
actually more interesting, because there the momentum is proportional
to the square root of the energy, but another factor of momentum
(or velocity) enters because higher-momentum molecules collide more
frequently. Photons all travel at the same speed, but then their
momenta are proportional to their energies already.
The real difference between the photon gas and a gas of nonrelativistic
molecules is that for the former, the number of particles per unit
volume is not conserved, but rather is determined by the temperature.
Dan Schroeder
dschroeder@cc.weber.edu
p.s. That last sentence is misleading: delete "per unit volume".