In one embodiment, a processor includes: a plurality of cores; a power controller including a logic to autonomously demote a first request for at least one core of the plurality of cores to enter a first low power state, to cause the at least one core to enter a second low power state, the first low power state a deeper low power state than the second low power state; and an interface to receive an input from a system software, the input including at least one demotion control parameter, where the logic is to autonomously demote the first request based at least in part on the at least one demotion control parameter. Other embodiments are described and claimed.