Does this refrigeration damage the bowl?
When I had one it was non stick aluminum which is something I'd happily put in the fridge. This seems to be common. The bowl is oversized for the amount of rice you'd want to chill, so it dries out. Anyway you should cool rice quickly, such as with cold water which would then need draining, at which point you might as well use a smaller container. I wouldn't usually have the fridge space for a rice cooker bowl.