No, they shouldn't. The north and south poles have vastly different eco systems. The north pole ice sits atop an ocean. The water beneath is being constantly warmed by ocean currents. Heat is constantly being transported into the area. Contrast that with the south pole. It sits atop a continent of solid rock. The rock can cool much more than liquid water. There is much less heat transported to Antarctica by ocean currents. Therefore the temperatures drop much lower.

It only takes about 3 seconds of thought to realize that it makes perfect sense that Antarctica is colder than the Arctic.