With increase in pressure (above atmospheric pressure), the C_p of a gas

The $${C_P}$$ increases because in case of higher pressure it has to do more work on the surroundings there by for on given amount of heat more amount of heat will be used to do work when compared to previous case where the pressure is slightly lower and hence more amount of heat has to be given to the system to raise the system to same temperature as of previous case, so the value of $${C_P}$$ increases.