Heat flow transects across the Norwegian-Barents-Svalbard continental margin (Sundvor and Eldholm, 1992) reveal anomalously high (by about 60 mK/m) sub-seafloor thermal gradients on the upper continental slope, at water depths about 600m--800m. Of several hypotheses which might explain these enhanced gradients, deep crustal fractures or igneous activity along the ocean continental crustal boundary (COB) is rejected by the lack of associated seismicity, extrusive/intrusive activity, or systematic correlation of gradient anomaly with the COB, which does not correspond to any specific water depth. A ''Memory in the Mud'' explanation is possible if the base of the Norwegian-Atlantic (warm) current rose by ca. 150--200 m ca. 50--300 years ago. However, high local spatial variability in observed gradients renders this mechanism suspect. Alternatively the thermal gradients were locally steepened by enhanced rise of pore fluids, mud and/or methane gas. This is most likely on the upper continental slope because post-opening sediment accumulations (i.e., dewatering) tend to be maximal there. ¿ American Geophysical Union 1996