<p>Low-density, selective grazers with restricted habitat requirements may be particularly vulnerable to extinction under hotter, drier conditions expected across Africa’s drylands with climate change. We evaluated how spatiotemporal variation in vegetation greenness and environmental heat load influenced movements of sable antelope, a water-dependent African grazer with selective resource requirements. We recorded hourly Global Positioning System (GPS) locations of 10 sable antelopes (<i>Hippotragus niger</i>) in Bwabwata National Park, Namibia, for up to 24 months, with concurrent hourly measurements of environmental heat load. As conditions became hotter and vegetation became browner during the dry season, home ranges increased in size and elongated towards the Kavango River. The 24-hour mean hourly displacement distance of sable antelopes increased in response to increased heat load and brown vegetation exposure, primarily due to more frequent long-distance movements to the Kavango River during the late dry season. Using Hidden Markov Models (HMMs), we identified four behavioural states of movement from GPS movement tracks: resting, foraging, local movement, and relocating. The relocating state, which was generally associated with directed movements to surface water, predominated during the late dry season. Sable antelopes increased both relocating and local movements, while decreasing foraging when exposed to high heat loads and brown vegetation. As conditions become increasingly hotter and drier, and resource availability becomes more unpredictable in space and time, species with specific resource requirements may face increased risk of extirpation, as walking farther to meet water requirements may compromise their already precarious energy balance during periods of resource limitation.</p>