Looking at the orbit of the moon which does figure-8s through the hole in the middle, isn't it conceivable that a sun can orbit that same way? Or rather, the two will orbit each other in a strange non-trivial way.
It would have to be a very small sun tough in order to fit trough the hole and not burn the planet to a crisp. So we arrived at the Discworld, where the elephants carrying the world sometimes have to lift their legs to let the sun pass.
Such a small star would probably have to be kept burning artificially, as the smallest mass still allowing nuclear fusion is around 80 times the mass of Jupiter.
The ringworld has a radius approximately that of the radius of the orbit of earth. A quite large sun could fit easily through such a 'toroidal planet.'
