Time-resolved elliptically polarized pump-probe spectroscopy developed by us is used to study spin relaxation dynamics of electrons in an intrinsic GaAs under an external magnetic field. Absorption quantum beats are observed and attributed to Larmor precession of electron spin around the magnetic field. The frequency of this quantum beat is used as a new method to measure electronic g factor with high accuracy. In this paper, this new method is utilized to investigate the temperature and energy dependence of electronic g factor in the intrinsic GaAs. It is found that electronic g factor increases with the temperature and energy of electrons, which is guitc different from the dependence predicted by k·p theory. An empirical formula is given to describe the temperature and energy dependence of g factor by fitting experimental data.