The physiological and biomechanical requirements of flight at high altitude have been the subject of much interest. Here, we uncover a steep relation between heart rate and wingbeat frequency (raised to the exponent 3.5) and estimated metabolic power and wingbeat frequency (exponent 7) of migratory bar-headed geese. Flight costs increase more rapidly than anticipated as air density declines, which overturns prevailing expectations that this species should maintain high-altitude flight when traversing the Himalayas. Instead, a "roller coaster" strategy, of tracking the underlying terrain and discarding large altitude gains only to recoup them later in the flight with occasional benefits from orographic lift, is shown to be energetically advantageous for flights over the Himalayas.