We propose a multi-symplectic generalisation of Souriau’s Lie group thermodynamics for first order parametrised classical field theories. A new notion of general covariant Gibbs state functional is defined in terms of the multi-momentum map associated to the lifted action of the diffeomorphisms group on the fields extended phase space. We elaborate on the use of such functional toward a covariant statistical mechanic description of fully constrained field theories, at the crossroad between geometrical methods and information theory.