The effect of anisotropic grain boundary energy on the kinetics of stressed grain growth is investigated by two-dimensional phase field simulations. The simulations are performed for a two-dimensional representative volume element of polycrystalline copper with elastic cubic symmetry under both uniaxial elongation and shear loading with elastic strains in the order of a few thousandths. The Read-Shockley relation for the misorientation-dependence and the cubic symmetry for the inclination-dependence of grain boundary energy are assumed. The simulation results have generally shown that the misorientation-dependent anisotropy, compared to the inclination-dependent anisotropy, significantly affects the evolution of microstructure and texture during stressed grain growth. Particularly, the misorientation-dependent anisotropy of grain boundary energy results in (i) the elongated morphology of grains in the evolved microstructure; (ii) the aggregation of grains with similar orientations in the evolved microstructure; (iii) the strengthening of lower-misorientation angle grain boundary segments; and hence (iv) decelerating the overall rate of stressed grain growth and texture evolution and (v) decelerating stress relaxation under constant-strain loading. These results are based on the assumption that the isotropic grain boundary energy is equal to the high angle grain boundary energy of the misorientation-dependent grain boundary energy and the inclination-dependent grain boundary energy varies sinusoidally around the isotropic grain boundary energy.