The bee entrances in trees around here are typically not very big; 3 inches. Assuming the au natural habits of bees are worth emulating, it seems a 3 to 4” entrance would not be a hindrance to your bees. That is just a pure guess on my part though! I have hives with entrances anywhere from 2 bee widths to slots about 15” wide. If you have enough bees around your entrance to keep it protected, I would probably open it up a bit wider, especially in this heat.
Potential problems with a bigger entrance are: robbing, wax moths, small hive beetle.