When animals select areas to occupy, decisions involve trade-offs between the fitness benefits of obtaining critical resources and minimizing costs of biotic and abiotic factors that constrain their use. These processes can be more dynamic and complex for species inhabiting desert environments, where highly variable spatial and temporal distribution of precipitation can create high intra- and inter-annual variability in forage conditions and water availability, and thermal constraints can differ significantly among seasons and diel periods. We examined resource selection in desert bighorn sheep (Ovis canadensis mexicana) in Cabeza Prieta National Wildlife Refuge, Arizona, USA, at multiple spatial and temporal scales to gain insight into how a desert mammal responds to variations in climatic conditions. We used resource selection functions to test topographic, forage, and environmental features among seasons and diel periods, and between non-drought and drought conditions at the population and home-range scale. When precipitation was average, sheep selected for topographic features that were beneficial for predator avoidance (i.e., escape terrain—steep, rugged areas with high visibility) and locations near perennial water. When drought occurred, they ranged further from preferred escape terrain and perennial water, perhaps seeking forage conditions suitable to meet their nutritional requirements. On early (April–June) and late (July–September) summer days, sheep selected for more northerly aspects and locations with lower solar radiation, and in some periods, selection for these cooler areas coincided with periods when forage covariates, proximity to perennial water, and several topographic features were uninformative in resource selection models. These choices may be necessary trade-offs, foregoing good escape terrain and foraging areas, and access to water, for improved thermoregulation. This study highlights the importance of identifying resource selection at variable spatial and temporal scales when investigating the interrelationship between species and their environment. It provides insight into the dynamics of resource selection in desert mammals, and how they respond to constraints imposed on them by their environment. This work can serve to inform strategies for managing and conserving species living in arid environments when faced with climate change.