Gibbs-Duhem equation is a fundamental relation of the thermodynamics of fluids that originates from the extensive behavior of fluids. Fluids adsorption in micro-porous media can break the extensivity with respect to volume, so that Gibbs-Duhem equation does not hold in general for adsorbed fluid. As a consequence, a total of 6 moduli are needed to fully describe the thermo-mechanics of an adsorbed fluid instead of 3 for bulk fluids (compressibility, thermal expansion, and heat capacity). In recent years, poromechanics theory has been extended to capture the effects of fluid adsorption in micro-porous media, but the non-validity of Gibbs-Duhem equation has been disregarded so far. We propose here a new formulation of poromechanics extended to adsorption which does not assume Gibbs-Duhem equation. The introduction of 6 fluid moduli instead of 3 is a rather fundamental change that requires to revisit the derivation from the fundamentals of poromechanics. In addition, the new poromechanics is formulated to capture thermo-mechanical couplings in double porosity media (micro- and macro-pores) with potential fluid transfers between the porosities. The final constitutive equations exhibit the same structure as in usual poromechanics, but with effective poromechanical properties that are affected by fluid confinement. The impact of assuming Gibbs-Duhem equation or not is discussed. In particular, we show that relaxing the assumption of Gibbs-Duhem may explain some experimental observations of adsorption-induced effects that are not captured by existing poromechanics. The proposed theory should improve our understanding of micro-porous materials with wide perspectives for applications (clay, cement, wood, bones, microporous carbons etc.).