The light we receive from the Sun depends on its temperature and its apparent size in the sky, but not much else. For example, the edges of the Sun don't seem bright, because even though we are looking at a much larger surface area, we're seeing it at an angle. If we were nearer to the Sun it would take up more size in the sky, but any individual part would seem the same brightness, just as a light bulb doesn't seem like it gets brighter if you closer to it.
Your torus model is a Sun with a hole, or a Sun that has the same radius in one direction but much less (unspecified) in another. So we get much less light. For reference, Mars is at 1.52 AU; the inverse square is 43%, and a little outside of the habitable zone. If your torus occupies less than half the apparent area of the Sun, your Earth will be in deep trouble. Something much less dramatic will still cause an Ice Age, at least - even esoteric variations in Earth's real orbit and axial tilt have been blamed for as much.
If your Sun is rotating very rapidly so that Doppler effect is relevant, this answer could change, but I'm not sure in what direction. I don't seriously have a model for a toroidal star. I'm also assuming the surface temperature (color) is the same as ours; if you make it a hotter star you can compensate, at the cost of a bluer sunset.