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Natural geomaterials such as fissured rocks and aggregated soils often exhibit a pore size distribution with two dominant pore scales, usually termed macropores and micropores. High-fidelity descriptions of these materials require an explicit treatment of the two pore regions as double porosity. In this paper, we present stabilized mixed finite elements for coupled flow and deformation in porous media with double porosity based on a thermodynamically consistent modeling framework. Mixed finite elements that interpolate the solid displacement and pore pressures in the macropores and micropores are used for this purpose. In the limit of undrained deformation, the two pore pressures become incompressibility constraints that cause an inf-sup instability in the form of spurious oscillation at the two pore scales. To circumvent this instability we develop a variant of the polynomial pressure projection technique for a twofold saddle point problem. The proposed stabilization allows the use of equal-order (linear) interpolations of the displacement and two pore pressure variables throughout the entire range of drainage condition.