N.H. Rev. Stat. Ann. § 365:34

Any public utility ordered to make reparation hereunder shall conform to said order and make payment as required therein. Upon failure to make such payment upon demand, the amount ordered paid may be recovered with interest in an action of debt brought by the person to whom payment was ordered; except that, in the case of costs ordered paid to the commission, suit shall be brought in the name of the state. Like action may also be brought to recover fees due the commission.