We applied two metrics, apparent temperature and humidex, to calculate heat stress in both present and future climates. We use an ensemble of CORDEX-Africa simulations to estimate heat stress during a baseline period and at two specific warming levels, 2 and 4 degrees C above pre-industrial. The increase in temperatures and changes to the precipitation distribution under climate change are projected to increase the intensity of heat stress events in Sahelian Africa and introduce new heat stress events in Northern and Central Africa. As the intensity of heat stress increases, it is expected that the use of energy-intensive cooling will increase. The energy system, therefore, will need to be able to supply more energy to power fans or air conditioning units. The cooling demand to turn a heat stress event into a non-heat stress event is computed. This value is then weighted by the population to find the total cooling required to prevent heat stress across the continent. Country-level results indicate that the greatest increases in cooling demand will occur in countries with densely populated regions, most notably Nigeria. Supplying this additional cooling demand will present the greatest challenge to less developed countries like Somalia. We find the least-cost future energy system that meets the projected increase in demand and derive the increase in energy system costs with the TIAM-UCL model. The total increase in energy costs to prevent heat stress is found to be $51bn by 2035 and $487bn by 2076.