Confession time: the "rum" I used was actually rum extract. I rarely buy rum, but when I do I don't use it for baking. ☺ For the record, I buy rum during the holidays to make hot buttered rums, and ultimately, those hot buttered rums served as the inspiration for this coffee cake. Sort of. At least they inspired me to use browned butter icing. Seeing as how it's June and not rum season, I decided to buy the rum extract. If you keep real rum on hand and want to use that then I say, "Go crazy nuts!!" I'm just not sure how much real rum you would use so you'll have to experiment. That' not a bad thing.
Note: I did not actually wait until the coffee cake was completely cooled before I added the icing. Consequently, the icing melted and was a little runny. I recommend that you wait for the cake to cool. If not, it's all good. It still tastes great and when you cut that first slice, the icing will run to the bottom of the pan.