Plot – Project RegionPlot3D to 2D with the Hue / Color function to display the third dimension

I want to project the output of RegionPlot3D to a 2D plane. Basically, I want to turn this around

f = a ^ 2 + b ^ 3 + c;
RegionPlot3D[f[f[f[f<= 0.5, {a, 0, 1}, {b, 0, 1}, {c, 0, 1}, 
 AxesLabel -> Automatic, ColorFunction -> "Rainbow",
ViewPoint -> Above, Mesh -> None]

rather into a contour representation (color that represents the value c). I am only interested in the maximum value for c, which meets the restrictions. Therefore I would like to project on a 2D-level.

My actual functions are much more complex and the 3D image makes it harder to see the details.