The ocean heat transport into the Arctic and the heat budget of the Barents Sea are analyzed in an ensemble of historical and future climate simulations performed with the global coupled climate model EC-Earth. The zonally integrated northward heat flux in the ocean at 70A degrees N is strongly enhanced and compensates for a reduction of its atmospheric counterpart in the twenty first century. Although an increase in the northward heat transport occurs through all of Fram Strait, Canadian Archipelago, Bering Strait and Barents Sea Opening, it is the latter which dominates the increase in ocean heat transport into the Arctic. Increased temperature of the northward transported Atlantic water masses are the main reason for the enhancement of the ocean heat transport. The natural variability in the heat transport into the Barents Sea is caused to the same extent by variations in temperature and volume transport. Large ocean heat transports lead to reduced ice and higher atmospheric temperature in the Barents Sea area and are related to the positive phase of the North Atlantic Oscillation. The net ocean heat transport into the Barents Sea grows until about year 2050. Thereafter, both heat and volume fluxes out of the Barents Sea through the section between Franz Josef Land and Novaya Zemlya are strongly enhanced and compensate for all further increase in the inflow through the Barents Sea Opening. Most of the heat transported by the ocean into the Barents Sea is passed to the atmosphere and contributes to warming of the atmosphere and Arctic temperature amplification. Latent and sensible heat fluxes are enhanced. Net surface long-wave and solar radiation are enhanced upward and downward, respectively and are almost compensating each other. We find that the changes in the surface heat fluxes are mainly caused by the vanishing sea ice in the twenty first century. The increasing ocean heat transport leads to enhanced bottom ice melt and to an extension of the area with bottom ice melt further northward. However, no indication for a substantial impact of the increased heat transport on ice melt in the Central Arctic is found. Most of the heat that is not passed to the atmosphere in the Barents Sea is stored in the Arctic intermediate layer of Atlantic water, which is increasingly pronounced in the twenty first century.