The zero-dimensional semiconductor nanostructures belong to the candidates for the realization of the quantum bits. They are expected to be scalable for the purpose of tuning their physical properties. In these structures the quantum bit could be realized in the form of a single quantum dot with two electronic energy levels, with only one electron in the dot. As the basic states of the quantum bit, realized in this way, the two orbital states of the electron in the dot could be used. It appears however that usually the relaxation of the energy of the electron from the excited energy level is often rather fast in the polar semiconductor quantum dots. It is the purpose of this paper to present calculations of the relaxation rate of the electron in an asymmetric pair of tunneling coupled quantum dots, in which the two electronic orbitals of the quantum bit are located each in a separate dot. The calculation of the electronic energy relaxation is based on the multiple electron-LO-phonon scattering processes, implemented to the theory via the electronic self-energy taken in the self-consistent Born approximation. The dependence of the relaxation rate on the geometry of the pair of the coupled dots and on the lattice temperature is presented for a realistic model of this nanostructure.