The brain's functional connectivity is complex, has high energetic cost, and requires efficient use of glucose, the brain's main energy source. It has been proposed that regions with a high degree of functional connectivity are energy efficient and can minimize consumption of glucose. However, the relationship between functional connectivity and energy consumption in the brain is poorly understood. To address this neglect, here we propose a simple model for the energy demands of brain functional connectivity, which we tested with positron emission tomography and MRI in 54 healthy volunteers at rest. Higher glucose metabolism was associated with proportionally larger MRI signal amplitudes, and a higher degree of connectivity was associated with nonlinear increases in metabolism, supporting our hypothesis for the energy efficiency of the connectivity hubs. Basal metabolism (in the absence of connectivity) accounted for 30% of brain glucose utilization, which suggests that the spontaneous brain activity accounts for 70% of the energy consumed by the brain. The energy efficiency of the connectivity hubs was higher for ventral precuneus, cerebellum, and subcortical hubs than for cortical hubs. The higher energy demands of brain communication that hinges upon higher connectivity could render brain hubs more vulnerable to deficits in energy delivery or utilization and help explain their sensitivity to neurodegenerative conditions, such as Alzheimer’s disease.