This is an important distinction so we are going to go ahead and underline it for clarity. Unless your cooler says specifically that it is bear proof, then it is not bear proof.
In addition, most coolers are actually easily accessible by common forest critters like raccoons, skunks, and even smaller rodents and squirrels. Some coolers have padlock loops available on the lid allowing you to lock it shut, but many others do not.
If you are worried about bears, you should watch How to Hang a Bear Bag.