So can all MPs get their energy bills paid for?

No – this is a distortion of a policy just for MPs who have to have two residences for their work.
Firstly, only MPs with constituencies outside of London are apply to claim any sort of utility bills back.
Secondly, they can only do so on one of the properties which they use, either the London one – which is required so they can be close to parliament – or their constituency home.

These costs include rental payments, utilities (gas, electricity, other fuel and water), council tax, round rent and service charges.
Buildings insurance, routine security measures, a landline telephone, a broadband connection and usage charges for that one property are all included, too.
And thirdly, any claims for utility bills are not capped at £3,400. There is actually no upper limit.

https://www.huffingtonpost.co.uk/amp...0b3a5341b02d1/