Avoiding hyperthermia entails considerable metabolic costs for endotherms. Such costs increase in warm conditions, when endotherms may trade food intake for cooler areas to avoid heat stress and maximize their energy balance. The need to reduce heat stress may involve the adoption of tactics affecting space use and foraging behaviour, which are important to understand and predict the effects of climate change and inform conservation. We used resource selection models to examine the behavioural response to heat stress in the Alpine ibex (Capra ibex), a cold-adapted endotherm particularly prone to overheating. Ibex avoided heat stress by selecting the space based on the maximum daily temperature rather than moving hourly to ‘surf the heat wave’, which minimised movement costs but prevented optimal foraging. By integrating these findings with new climate forecasts, we predict that rising temperatures will force mountain ungulates to move upward and overcrowd thermal refugia with reduced carrying capacity. Our approach helps in identifying priority areas for the conservation of mountain species.