Gallium nitride crystals with their comparatively high-energy phonons, and strong electron-phonon coupling, seem to be promising for the phonon band population inversion affecting the two-phonon-difference absorption of infrared radiation. We devote this work to the theory of phonon generation by electrons in n-type GaN crystals at the lattice temperature T between 300 and 10 K, including deformation- and piezo-potentials, as well as ionized impurity contributions to the scattering rate of electrons. We determine the non-equilibrium phonon numbers using our calculated Monte Carlo data on phonon generation rate and literature data on phonon lifetimes. We conclude the inelastic process of optic phonon emission by electrons to be dominating in the electric field range from 1 to10 kV/cm, and the optic phonon band population to be much in excess compared to the acoustic one at the same wave vector values. We give a trial of the phonon-photon coupling constant values for the two-phonon-difference process together with the non-equilibrium phonon number values, including photon re-absorption by free carriers, to calculate the resulting absorption coefficient. It turns out to be higly sensitive to electric fields in the range from 1 to 10 kV/cm, in the photon frequency range of 15-20 THz.