2 tips I remember from when my kids had hamsters are to stand the cage on a cold surface and also to wet a towel with very cold water, wring it out tightly and put it over the cage for a while.
Don't put him in the fridge!! I know you were only joking, but if you put him in the fridge he'll end up hibernating.
Don't forget hamsters come from semi-desert areas, so they are well adapted to coping with the heat.
You could try filling a deep tray with earth, quite well compacted, like it would be in the garden, and then he can dig himself a burrow, like he would in the wild. The earth will help to keep him cool.
Give him cucumber and lettuce, to keep him hydrated.