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Actions usually taken to prevent processors from overheating, such as decreasing the frequency or stopping the execution flow, also degrade performance. Multiprocessor systems, however, offer the possibility of moving the task that caused a CPU to overheat away to some other, cooler CPU, so throttling becomes only a last resort taken if all of a system's processors are hot. Additionally, the scheduler can take advantage of the energy characteristics of individual tasks, and distribute hot tasks as well as cool tasks evenly among all CPUs.This work presents a mechanism for determining the energy characteristics of tasks by means of event monitoring counters, and an energy-aware scheduling policy that strives to assign tasks to CPUs in a way that avoids overheating individual CPUs. Our evaluations show that the benefit of avoiding throttling outweighs the overhead of additional task migrations, and that energy-aware scheduling in many cases increases the system's throughput.