The fire broke out at about 22:45 GMT on Friday at West Cumberland Hospital, destroying the energy building which will supply the hospital's power.
North Cumbria University Hospitals NHS Trust said it was "unclear" how the blaze started. No-one was hurt.
Chief executive Ann Farrar said there would be a "knock on effect" for the opening of the hospital in April which was "absolutely devastating" news.
No patients were affected by the fire and all services continue to run as usual, the trust said.
Labour MP for Copeland Jamie Reed said everything possible must be done to "facilitate the quickest opening".
He said: "It's now imperative that everyone involved with our local health economy pulls together in the same direction to ensure that the opening of the new West Cumberland Hospital and the transfer of staff and clinical services is achieved as quickly as possible.
"This is a sickening disappointment for the whole community, but the new hospital will open, we will get there."
Cumbria Fire and Rescue Service said five fire engines and 30 firefighters tackled the blaze which took about four hours to put out.
An investigation is under way.
Part of a new £90m hospital in Whitehaven has been damaged in a fire.