Definitely more cost efficient
I would be a bit worried about putting an ice pack in the cage to be honest. Anything ceramic or terracotta of any shape is fine. When we had a heatwave one year I had two ceramic mugs I used and would swap them round - kept one in the fridge. When the one in the cage warmed up too much I'd swap it with the one in the fridge and put the other one in the fridge.
Teracotta is quite good as it's absorbent and you can just sprinkle a tiny bit of cold water on it to cool it down and it's soaked up.
We don't often get heatwaves in the Uk! Or for long - so it's not an ongoing thing
A fan in the room can help as well.