We study analytically and numerically the thermoelectric properties of a chain of cold atoms with dipole-dipole interactions placed in an optical periodic potential. At small potential amplitudes the chain slides freely that corresponds to the Kolmogorov-Arnold-Moser phase of integrable curves of a symplectic map. Above a certain critical amplitude the chain is pinned by the lattice being in the cantori Aubry phase. We show that the Aubry phase is characterized by exceptional thermoelectric properties with the figure of merit ZT = 25 being 10 times larger than the maximal value reached in material science experiments. We show that this system is well accessible for magneto-dipole cold atom experiments that opens new prospects for investigations of thermoelectricity.