The expenditure of energy involves elaborate chemical reactions, most of which are sensitive to temperature. Birds have a somewhat higher metabolic rate than mammals, but not significantly so. Birds and mammals are similar in their metabolic adaptations, but there are differences as well 1. Distribution of fat in birds bodies is different for aerodynamics.
Birds have no sweat glands and lose heat through their respiratory system and exposed skin.
Birds have to incubate eggs outside their body, requiring additional heat. The energy cost of activities above the basic level seems to be no more in birds than in mammals. But the behavior of birds tends to make them engage in more energy-costly activities. They are diurnal, while mammals are nocturnal and thus are more exposed to wind, temperature fluctuation, and predators. The smaller the bird, the more the energy used per gram of body weight because the increase in energy need doesn't increase quite as fast as weight.