Get a meter from your dealer or buy your own, to put on the tub for 6 months, preferably all extremes. This is the only true way to know what the tub is costing. There can be many factors that are effecting your increase, more dryer time for towels, more lights for entrance and exit from the tub, amount of use, climate variations to name a few of many possible causes.
Do you use a floating blanket under your hard cover? Is the hard cover fitting properly and always down tight? How much do you use the tub? Have your kwh rates increased because of a tier system? How often do you change your water? How is your filtering programmed? Double per month means how much to us? From 30-60 or more like from 100-200?
I could go on and on but there are some things to start checking into.