The contour is an approximation of the raster. Wouldn't it be easier to calculate the areas directly by binning the raster data?
Anyway, according to the documentation for rasterToContour, the results is a SpatialLinesDataFrame (sp package). So you should be able to use functions from the sp package to calculate the areas. You might need to convert the spatial lines to polygons first, and then maybe intersect the polygons for the different contours.