radiation pressure

The trace of the energy momentum tensor is zero for a photon fluid, \(g^{\mu\nu}T_{\mu\nu} = 0\), and direct evaluation shows that it is at the same time equal to \(\rho - 3p = 0\), implying \(w = p/\rho = 1/3\) in three dimensions. Can you generalise this statement to \(n\) dimensions? Why, intuitively speaking, is the equation of state parameter smaller for spaces with more dimensions, or equivalently, why is there less radiation pressure?