It's apparent that the immune system, while fighting an infection, needs some energy, and some folks have tried to measure how much energy exactly. The figures from different sources are:
- Between 4 and 7 kcal/kg per day. source.
- 14 percent increase in resting rate for respiratory tract infections without fever. Between 7 and 15 pecent increase for every extra °C. source
- Up to 55 percent increase for serious diseases. source
This is surely a lot.