  We rewrite the Klein-Gordon (KG) equation for a complex scalar field as a new
Gross-Pitaevskii (GP)-like equation. The potential of the scalar field is a
mexican-hat potential and the field is in a thermal bath with one loop
contribution. We interpret the new GP equation as a finite temperature
generalization of the GP equation for a charged field. We find its hydrodynamic
version as well and using it, we derive the corresponding thermodynamics. We
also obtain a generalized first law for a charged Bose-Einstein Condensate
(BEC).
