You're probably burning more energy turning it off and on again. It doesn't really use any noticeable power sitting idle.
Anyway, a direct comparison would be pretty difficult because your model is probably tens of billions of parameters, not over a trillion. Energy consumption per output token will probably be a bit higher for the frontier models but something that people have found is that higher quality models often need fewer tokens to achieve the same goal. Plus how many times do you re-prompt your local model vs Claude Fable or Opus for example to get the desired result?