Aiming at problems of low energy storage efficiency and unstable energy output of existing accumulators, this paper proposes a novel constant pressure elastic strain energy accumulator based on the rubber material hyperelastic effect, which can store and release energy with steady constant pressure. Based on exergy analysis method, constant pressure elastic strain energy accumulator charging/discharging energy storage efficiency is analyzed. Then Mullins effect on the rubber airbag multiple charging/discharging cycles is studied. Finally, a test platform is set up to verify the energy storage efficiency, expansion and contraction pressure stability of the rubber accumulator during charging/discharging cycles. Compared with enthalpy analysis method, experiment results show that energy storage efficiency calculation by the exergy analysis method is more accurate. In more than 200 cycle tests, rubber airbag energy storage efficiency is always higher than 76%, and expansion pressure and contraction pressure errors under steady state are less than 2.92e-3MPa and 1.79e-3MPa, respectively. The results show that the rubber airbag can be used as an effective energy storage component, which is very meaningful for energy recovery in pneumatic or hydraulic systems.