While the curvature of the classical liquid surfaces exhibits only a weak temperature dependence, we demonstrate here a reversible temperature-tunable concave-convex shape switching in capillary-contained, surfactant-decorated, oil-water interfaces. The observed switching gives rise to a concave-convex shape transition, which takes place as a function of the width of the containing capillary. This apparent violation of Young's equation results from a hitherto-unreported sharp reversible hydrophobic-hydrophilic transition of the glass capillary walls. The transition is driven by the interfacial freezing effect, which controls the balance between the competing surfactants' adsorption on, and consequent hydrophobization of, the capillary walls and their incorporation into the interfacially frozen monolayer. Since capillary wetting by surfactant solutions is fundamental for a wide range of technologies and natural phenomena, the present observations have important implications in many fields, from fluid engineering to biology, and beyond.