How often can you refreeze the same cake?

You should avoid re-freezing anything that has been frozen and thawed. Serious gastric upsets could result.

A different perspective . . .Generally, the main concern with a cake (depends on the cake, of course) would not be food safety, but the extra freeze-thaw cycle would definitely affect the quality of the cake - making it dry out, become crumbly or otherwise making it less presentable.