In the present paper, we study a linear thermoelastic porous material with a constitutive equation for heat flux with memory. An approximated theory of thermodynamics is presented for this model and a maximum pseudofree energy is determined. We use this energy to study the spatial behavior of the thermodynamic processes in porous materials. We obtain the domain-of-influence theorem and establish the spatial decay estimates inside of the domain of influence. Furthermore, we prove a uniqueness theorem valid for finite or infinite bodies. The body is free of any kind of a priori assumptions concerning the behavior of solutions at infinity.