What temperature does chicken breasts need to be before putting them in the fridge?
After they're cooked? I generally let them cool a bit at room temperature, then stick them in the fridge in a container with the lid cracked open, preferably with the breasts in a single layer so they cool more quickly. Once they've gotten cold I shut the lid.
With any food, you have 2 hours to cool it to 70 degrees, then 4 hours to cool it to 40 degrees or below. The critical part is the 70 degrees. Once it reaches that, you can refrigerate it and trust that it will reach 40 or below within 4 hours.