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That would depend on the cooler specification, if the cooler meets the processor requirements there should not be any difference; however, getting a good custom cooler can get you special features that a cheaper one cannot provide since it only meets the minimum requirements. If you have the option to get the good custom cooler, I definitively recommend it as long as it meets the processors requirements.
Thank you for the very informative reply.
Helps a lot.
Yeah upgrading from the 3770k. Had it since 2012 when it released.