This study presents a thermodynamic analysis of an ideal integrated adiabatic compressed air energy storage (A-CAES) system with single-temperature thermal energy storage (TES) subjected to an external heat boost. The additional heat is assumed to be supplied by concentrated solar energy, although other heat sources, such as combustion, can also be used, at least in principle. We examine three principal performance metrics of the integrated A-CAES system: the round-trip coefficient of performance (CoP), the marginal thermal CoP of the solar booster, and the second-law efficiency of the overall integrated system. The principal thermodynamic constraint is formulated as a theorem for the upper bound of the integrated 2nd law efficiency. The analysis shows that the system can attain a high round-trip CoP while simultaneously utilising the external solar heat boost very efficiently. These results highlight the thermodynamic potential of integrating solar heat boosting with A-CAES to improve both storage performance and renewable heat utilisation.