The Barents Sea is a primary gateway for Atlantic Water entering the Central Arctic Ocean and ubiquitous water-mass transformation on the Barents shelf is key for mitigating increases in heat transport to the central Arctic through the St. Anna Trough. Using a mesoscale-permitting reanalysis spanning 40 years, we derive the first long-term estimate of heat transport through the St. Anna trough, finding that it has increased by 0.15 TW/year since 1980. However, this is only half of the 0.28 TW/year trend in increasing heat transport into the Barents Sea through the Barents Sea Opening. Decomposing the heat transports reveals that these trends are entirely due to warming temperatures at the sections with no discernible trend in the volume transports. We find that a northward migration of the largest heat fluxes from the ocean to the atmosphere have resulted in cooler and denser Northern Barents Shelf Water, mitigating the heat transported through the St. Anna trough. However, despite functioning properly, the "Barents Cooling Machine" has been unable to keep pace with the dramatic warming of the Atlantic Water inflow, resulting in the residual trend in heat transport to the central Arctic. Finally, we present the first observational evidence for the “ocean feedback” hypothesis, indicating that it modulates volume transport into and out of the Barents Sea on sub-decadal timescales.