Several studies conducted on cooling methods of the supercritical carbon dioxide Brayton recompression cycles operating in arid regions suggest that hybrid cooling could be more advantageous than wet and dry cooling due to the challenges experienced in the regions. As an alternative to traditional cooling technologies, this project investigates the possibility of using a vapor compression refrigeration cycle for heat rejection for a supercritical carbon dioxide (s-CO₂) Brayton Recompression cycle (BRC). Previous studies have shown that maintaining the main compressor inlet temperature (CIT) close to critical point temperature could potentially increase the overall cycle efficiency by up to 5%. Since arid regions are known for high ambient temperatures for most of the year, achieving the recommended CIT would require the implementation of some of the innovative cooling technologies. This study focuses on the use of a vapor compression refrigeration cycle to effectively cool the s-CO₂ BRC. The study capitalizes on the refrigeration cycle's ability to reject heat at higher temperature environments, which would be of great benefit to ensure a low and stable CIT, even at the high ambient temperatures. This study targets CIT ranging between 30°C and 35°C. Previous research has shown that CIT ranging in the above bracket is essential to achieving higher cycle efficiencies, especially at a relatively high compression ratio. To achieve the above, a mathematical model of the complete cycle is developed, and a simulation model is built in Flownex simulation environment to predict the cycle overall response assuming constant heat input and variable ambient temperature. Results show that the vapor compression refrigeration cycle has the potential to successfully cool the supercritical carbon dioxide Brayton recompression cycle under the above-prescribed conditions, recommending further research into overall cycle efficiency enhancement.