A BOD protects the microcontroller when the supply voltage falls below its operating threshold by resetting the device. This keeps the microcontroller in a defined state when the VCC is below its operating threshold. The BOD is not important to the microcontroller while it is in sleep mode but it is extremely important when it wakes up. Therefore, as a rule, most microcontrollers keep the BOD active during sleep mode and it contributes substantially to sleep mode power consumption.
The solution to this problem is to have the microcontroller shut down the BOD when it enters sleep mode and starts it again just before leaving sleep mode. This approach ensures the BOD is functioning when it is needed without any power consumption penalty while in sleep mode. However, this approach does add to the start-up time of the device, as the BOD needs additional start-up time.