We show that any universal curvature identity which holds in the Riemannian setting extends naturally to the pseudo-Riemannian setting. Thus the Euh-Park-Sekigawa identity also holds for pseudo-Riemannian manifolds. We study the Euler-Lagrange equations associated to the Chern-Gauss-Bonnet formula and show that as in the Riemannian setting, they are given solely in terms of curvature (and not in terms of covariant derivatives of curvature) even in the pseudo-Riemannian setting.
Gilkey, P., Park, J. H., & Sekigawa, K. (2012). Universal curvature identities II. Journal of Geometry and Physics, 62(4), 814–825. https://doi.org/10.1016/j.geomphys.2012.01.002