Recently, I have been interested in Wald's formula of entropy. In this formalism, there is a tensor $P^{abcd}=\frac{\partial L}{\partial R_{abcd}}$.
Among some papers they mention $P^{abcd}$, describing its symmetric anti-symmetric properties. But I want to know the explicit form of $P$ in certain cases.
There are known results from Einstein-Hilbert action (they mention, $P^{abcd}=\frac{1}{2}(g^{ac} g^{bd} - g^{ad} g^{bc})$ for Einstein Hilbert case), thus I assume that \begin{align} &\frac{\partial R}{\partial R_{abcd}}=\frac{1}{2}(g^{ac} g^{bd} - g^{ad} g^{bc})(?) \end{align} where $g$ is the usual symmetric metric, $R_{abcd}$ is Riemann curvature tensor, and $R$ is Ricci scalar.
It seems that they treat $R_{abcd}$ and $g_{ab}$ independently, so my first trial is decompose $R=g^{ac}g^{bd}R_{abcd}$, and try to compute $ \frac{\partial R_{pqrs}}{\partial R_{abcd}}$, from the symmetric arguments \begin{align} \frac{\partial R_{abcd}}{\partial R_{pqrs}}=\delta_{ab}^{pq}\delta^{rs}_{cd}+\delta_{cd}^{pq}\delta^{rs}_{ab} \end{align} But plugging this to $R$, I obtain a somewhat different answer shown above.
Am I doing something wrong?
If you have experienced this kind of derivative, do you have any hints or advice for this kind of algebraic computation?