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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 T4
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