If bear boxes or poles exist at your camp site, that would be the best choice when used properly, as David mentions above.
Many roto-molded (like Yeti, Orca, etc) coolers are IGBC approved too, which means they were tested with bears. You can follow the link below to a list of approved products, you might very well have one already:
Although for me, its the other critters that can be the most annoying! People I’ve camped with have had bags that were strung up invaded by mice and squirrels, they are crafty little buggers. I’ve watched a mouse find a forgotten wrapper in a backpack and chew through pockets to get it, and then a squirrel eat a bag of peanut M&M’s while helplessly watching 10 or more feet below. Ravens are pretty good at operating zippers - they will steal stuff from your backpack before you know it. Keeping camp clean and cooking away from camp are very important to keep all critters away!