Using extensive numerical and some exact asymptotic analyses, we argue that the canonical ensemble relaxation of a classical particle in an anharmonic multi-well potential landscape with leading quartic anharmonicity exhibits a 1/t decay to equilibrium. In addition, we provide numerical evidence of algebraic decay of form 1/tθ, 0 < θ < 1, for stronger leading anharmonicities. Our results have important consequences for relaxation studies in several problems of considerable interest in condensed matter physics such as in the Krumhansl-Schrieffer model of displacive structural phase transitions, and in various complex systems.