When the air temperature reaches 600K or higher, vibration is excited. The specific heat is not a constant but a function of temperature. Under this condition, the transition position of hypersonic sharp wedge boundary layer is predicted by using the improved eN method considering variable specific heat. The transition positions with different Mach numbers of oncoming flow, half wedge angles, and wall conditions are computed. The results show that for the same oncoming flow condition and wall condition, the transition positions of hypersonic sharp wedge boundary layer move much nearer to the leading edge than those of the flat plate. The greater the oncoming flow Mach number, the closer the transition position to the leading edge.