The thermal conductivity of a one-dimensional chain of rotators with a double-barrier interaction potential of nearest neighbors has been studied numerically. We show that the height of the "internal" barrier, which separates topologically nonequivalent degenerate states, significantly affects the temperature dependence of the heat conductivity of the system. The small height of this barrier leads to the dominant contribution of the non-linear normal modes at low temperatures. In such a case the coefficient of thermal conductivity turns out to be the risen function of the temperature. The growth of the coefficient is limited by local fluctuations corresponding to jumps over the barriers. At higher values of the internal barrier height, dependence of the heat conductivity on temperature is similar to that of classical rotators.