N.H. Rev. Stat. Ann. § 669:35

Any person aggrieved by a ruling of the board of recount with respect to any ballot may, within 5 days thereafter, appeal to the superior court for the county in which such town is located; and such court shall have jurisdiction in equity to hear and determine the question presented. The person shall also file a copy of the appeal with the town clerk by 5:00 p.m. on the fifth day following the date of the recount.