Can I use Brown sugar or Cane Sugar as a Substitute?

Cane sugar was just a source specific type of white sugar. so you can use that anyplace you see the ingredient listed as simply sugar. Some packages labeled simply sugar are actually made from sugar cane, while others are from sugar beets, or in a few instances it's a blend of the two. Chemically speaking, it's the same stuff. Some cane sugar is a bit darker, depending on the amount of molasses which was rinsed off. White sugar simply was washed more. The amount of molasses remaining is pretty small really, which is why you can sub one for the other without any real problems. Brown sugar, on the other hand, does have enough left behind to be problematic in baking. So I recommend you use that cane sugar.

I've made a very similar cake recipe that used all brown sugar. I think either will probably be fine!