Since the first connections between Feynman integrals and number theory were observed in the 1990's, a huge effort has gone into developing mathematics to better understand perturbative quantum field theory. Nonetheless, it is still an open question whether Feynman integrals admit a direct geometric interpretation within pure mathematics. In this talk I shall consider stable cohomology classes on real, complex, and quaternionic general linear groups, and show that their integrals over moduli spaces of tropical curves provide a class of Feynman integrals in 2 and 4 spacetime dimensions with remarkable properties. In particular, they are always finite.