Applications of (generalizations of the Chern-) Gauss-Bonnet theorem came up in the context of alternatives to General Relativity around fifteen to twenty years ago. You should probably stop reading here. :-)
As briefly touched on there, Einstein-Gauss-Bonnet (EGB) gravity turned out to require higher dimensions (the start of this abstract is worth the click :-) <https://link.springer.com/article/10.1140/epjc/s10052-020-82...> ) so the idea that EGB might sweep up higher order curvature in the self-interaction of physical gravitation is mostly dead.
Mostly. There's still a trickle in the literature. For example, <https://doi.org/10.1103/PhysRevD.104.044026> ("General relativity from Einstein-Gauss-Bonnet gravity", 2021, open access; ref [12] therein corresponds to <https://arxiv.org/abs/0805.3575> (no singularity in EGB black holes -- lots of Ricci calculus there, nothing like soccer balls as far as I can see)). The references in <https://arxiv.org/abs/1904.00260v1> (no singularity in Ef(G)B cosmology, applying a correcting function on the Gauss-Bonnet term) can takes an interested reader to years of discussions about the accelerated expansion of the universe. As always the first two digits of an arXiv identifier are the submission year.
https://en.wikipedia.org/wiki/Chern%E2%80%93Gauss%E2%80%93Bo... (the History section is pretty interesting)
https://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_gravity
As briefly touched on there, Einstein-Gauss-Bonnet (EGB) gravity turned out to require higher dimensions (the start of this abstract is worth the click :-) <https://link.springer.com/article/10.1140/epjc/s10052-020-82...> ) so the idea that EGB might sweep up higher order curvature in the self-interaction of physical gravitation is mostly dead.
Mostly. There's still a trickle in the literature. For example, <https://doi.org/10.1103/PhysRevD.104.044026> ("General relativity from Einstein-Gauss-Bonnet gravity", 2021, open access; ref [12] therein corresponds to <https://arxiv.org/abs/0805.3575> (no singularity in EGB black holes -- lots of Ricci calculus there, nothing like soccer balls as far as I can see)). The references in <https://arxiv.org/abs/1904.00260v1> (no singularity in Ef(G)B cosmology, applying a correcting function on the Gauss-Bonnet term) can takes an interested reader to years of discussions about the accelerated expansion of the universe. As always the first two digits of an arXiv identifier are the submission year.