Corten steel tends to overheat in hot environments? (for example copper heats up under the sun)
Bill summed it up pretty quick and provided a link. For some reason you are focusing on the copper content when it really is insignificant in the overall picture of the sun hitting its surface. If you could elaborate more, maybe we could offer a better answer.
Weathering steel, best-known under the trademark COR-TEN steel and sometimes written without the hyphen as Corten steel, is a group of steel alloys which were developed to obviate the need for painting, and form a stable rust-like appearance if exposed to the weather for several years. Overheating ? It's not possible for a metal to get hotter than it's environment unless there is a source of radiation. The sun shining on a metal will heat up a bit, as will any material, but the amount is small and depends on the surface reflectivity. Define what you mean by overheating. .