TobyC, you're absolutely correct! And the derivation in Wikipedia is not. Try this:
dU = T dS  P dV
(∂U/∂V)_{T} = T (∂S/∂V)_{T}  P
We need a Maxwell relation:
A = U  TS
dA =  S dT  P dV
⇒ (∂S/∂V)_{T} = (∂P/∂T)_{V}
Putting that in:
(∂U/∂V)_{T} = T (∂P/∂T)_{V}  P
Let u = U/V be the energy density. Then P = u/3 and ∂U/∂V = u gives you a differential equation
u = T/3 du/dT  u/3
whose solution is u = C T^{4}
