This paper presents a novel pressure swing adsorption process and the development of specifically designed sorbents for the process. It is operated at high temperature (650-800 degrees C) using the reversible reaction of calcium oxide with CO2, i.e., CaO + CO2 reversible arrow CaCO3. The new process directly stores the reaction heat released from the forward reaction in the sorbent and then releases it for sorbent regeneration under reduced CO2 partial pressure, so that the need of pure oxygen for oxy-fuel combustion is avoided. Two potential problems of the new process, namely, loss in capacity and slow and unmatched reaction rates of chemical-controlled carbonation and calcination, were discussed in detail. Three specifically designed calcium-based sorbents showed stable performance during 92 isothermal carbonation calcination cycles at either 680 or 750 degrees C. The calcination rate was significantly enhanced by increasing the reaction temperature and the introduction of steam to match the reaction rate of chemical-controlled carbonation. This pressure swing adsorption process could be used for low-cost CO2 separation using specifically designed sorbents under carefully selected operating conditions.