The answer doesn't lie within my particular area of expertise?, but I did find an interesting article, just below, that seems to say that appeals do run up the cost but the outlay begins even before that point. The cost of trials where the death penalty is being sought appears to be the starting point, i.e., when the defendant's life is on the line, the time spent in preparation ($$$), the additional experts ($$$), etc., all tend to rise dramatically.
Hope this helps (I haven't "vetted" - LOL - the article):