It can work, but please remember if it's airtight (even relatively so) that you really need to invest in a magnetic door that pops open or an alarm that calls you (and preferably a couple of other people nearby) in the event that the power goes out or the temperature rises dramatically (a malfunctioning cooler unit can blow hot air), or be absolutely sure someone can check it at least every hour. I've lost count of the number of cattle (that I know of) that have died in makeshift cooler rooms in recent years, just within say a 50 mile radius of me. Even because someone in a hurry put them in one morning and forgot to turn the cooler on...
We use a water cooler and fans under an open barn with shade cloth around the "cooling area". Not real cold, but it helps, and you can grow hair with a little work. It's just our choice, but at least we don't have to worry about killing our cattle if something goes wrong.