I know there's a paint for stoves, but I don't know for sure what the cost is. Maybe it's by rustoleum.
You may be able to put some type of protection on it. I would imagine, no matter what, it will rust some. Go to one of the hardware stores and talk to them about it.

There is a paint but not rustoleam.It is especially for very high heat for wood burning stoves.You can get it at Home DePot.Sand all the rust areas,clean the areas well and paint.It comes in spray and cans to be brushed on.I have used both.
moonphase

I believe it is called bar b Q black Used it up north on an indoor wood stove,yes even indoors they get rusty.