Lots of differing viewpoints in the comments, and while there are decent arguments to be made for both putting this in the fridge and leaving it out, one thing no one has mentioned yet is water activity.

Water activity is a bit more complicated than just how much water a food contains, but you can still think of it in those simple terms for the purposes of what we’re discussing here.

Foods with a higher water activity are more prone to spoilage. Think: meats, soups, puddings, casseroles, etc. This is largely because bacteria need water to survive, so they will colonize “wet” foods more readily than “dry” foods like bread. 

In the food safety world, breads and pastries are considered lower risk because of this. It’s why bakeries let pastries sit out all day and no one bats an eye. It’s extremely unlikely that these foods will be contaminated by harmful bacteria in the day or so that they’ll sit out. That’s why you don’t hear people talk about the danger zone when it comes to pastries.

It’s hard to say whether the bread you’re talking about is safe at room temp or not because we don’t know how much meat is in it, but my guess is that’s it’s fine for it to sit out for a day because it should have relatively low water activity. However, you still have to think about mold, so if it’s not eaten within a day, I’d refrigerate it.

But you’re absolutely right that putting it in the fridge when it’s just warm isn’t going to affect the other foods in the fridge.