  We obtain a new solution of the Einstein-anti-Maxwell theory with
cosmological constant, called anti-Reissner-Nordstrom-(A)de Sitter
(anti-RN-(A)dS) solution. The basic properties of this solution is reviewed.
Its thermodynamics is consistently established, with the extreme cases and
phase transitions, where the analysis is performed through two methods, the
usual one and that of Geometrothermodynamics . The Geometrothermodynamics
analysis does not provide a result in agreement with the usual method or by the
specific heat. We establish local and global thermodynamic stabilities of
anti-RN-AdS solution through the specific heat and the canonical and
grand-canonical ensembles.
