It depends on the type of film and on your post processing.
For black and white films there is not deed to cool them at all. When they mature well beyond their expiration date, they might get a bit slower if at all.
It is different for colour emulsions. The three or four colour "layers" may mature at different speed which may then result in unwanted colour shift, while a minor change to the overall speed of the film may not matter that much, similar to b&w.
For colour negatives, you could still correct minor colour shifts with your enlarger or have that automatically done in the lab where it is typically done for free anyway.
For slides it is different though. A slide is a slide with no chance of corrections due to the absence of any post processing. (Unless you plan to scan them or enlarge them on reverse paper or so.) The slide itself cannot be corrected in therms of speed (density) and colour. Slides should be kept cold when stored for some years.
And then there are the pro films. Pro films are typically used when ever very constant results in ever repeating processes are required. Such as Portraits in Photo Booth etc., where the operator is not even necessarily well trained. In order to keep constantly reliable results it is strongly recommended storing them at constantly cool temperatures, preferably in dry environments. No need to freeze them either.