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$S^0_{EH} = \int_M S(g) d\mu_g \rightarrow S_{EH} = \int_M \frac{S(g)}{vol(M)} d\mu_g$
So now consider adding only electromagnetic matter terms.  Suppose we accept the standard account of electromagnetism via connection $A$ as a connection on a $U(1)$ bundle with curvature $F=dA$.  Maxwell’s equation with source is $d *_g F = J$.  Now since we are on a 4-manifold, the Hodge $*_g$-operator is conformally invariant.  This conformal invariance is quite useful because this means the electromagnetic part of the Lagrangian may not care about scaling of the metric by $1/vol(M)$ which is a function of $g$