The Barents Sea is a major route for Atlantic Water to enter the Arctic. Cold air cools incoming Atlantic Water before it exits to the Arctic through the St. Anna Trough. We derive the first long-term estimate of the heat leaving the Barents Sea through St. Anna Trough. The heat leaving has increased since 1980, but only by half as much as the increase in heat entering. Finally, we present evidence for a previously proposed "ocean feedback" mechanism to help cool inflowing Atlantic Water.