A new scheme is investigated for evaluating the temperature dependence and dispersion relation of the Kerr constant (K) of an optically isotropic medium in isotropic and blue phases (BPs) liquid crystals. The scheme employs the measurement of the component of the transmitted light intensity of double modulated frequency using the modified in-plane-switching cell geometry (based on metallic film electrodes). It overcomes to a large extent the problem of a nonuniform electric field, employs relatively small driving voltages, and allows K to be measured directly. It is shown that the dispersion relation based on the single-band birefringence model describes well both blue and isotropic liquid crystal phases. It is found that the experimental data indicate that the temperature-dependent coefficients in this relation have a simple linear form in the isotropic phase, which allows a general model for the temperature and wavelength dependence of the Kerr constant in the isotropic liquid crystal phase to be formulated. In the BPs the temperature dependence of the experimental data deviate from the simple linear trend, but follow well an inverse exponential form.