Processor performance is fairly rigidly fixed to clock speed. It's possible that they make mistakes, catch the error and rectify it, in which case cooler will mean fewer errors and a slight net speed increase. I'm guessing there though.
Cooler processors suffer less from leak current and so use less electricity. Oddly as a result they also throw out less waste heat, so are easier to keep cold.
The main benefit is overclocking, where if "error correcting code" of some sort exists it's likely working overtime, leak currents are a bit ridiculous and power draw goes up sharpish, thus killing both previous arguments for keeping it cold.
Anyone know if processors make and correct mistakes. ecc ram style or otherwise?