Mixed matrix membranes (MMMs) consisting of a blend of a hydroxypolyamide (HPA) matrix and variable loads of a porous polymer network (PPN) were thermally treated to induce the transformation of HPA to polybenzoxazole (β-TR-PBO). Here, the HPA matrix was a hydroxypolyamide having two hexafluoropropyilidene moieties, 6FCl-APAF, while the PPN was prepared by reacting triptycene (TRP) and trifluoroacetophenone (TFAP) in a superacid solution. The most probable size of the PPN particles was 75 nm with quite large distributions. The resulting membranes were analyzed by SEM and AFM. Up to 30% PPN loads, both SEM and AFM images confirmed quite planar surfaces, at low scale, with limited roughness. Membranes with high hydrogen permeability and good selectivity for the gas pairs H2/CH4 and H2/N2 were obtained. For H2/CO2, selectivity almost vanished after thermal rearrangement. In all cases, their hydrogen permeability increased with increasing loads of PPN until around 30% PPN with ulterior fairly abrupt decreasing of permeability for all gases studied. Thermal rearrangement of the MMMs resulted in higher permeabilities but lower selectivities. For all the membranes and gas pairs studied, the balance of permeability vs. selectivity surpassed the 1991 Robeson’s upper bound, and approached or even exceeded the 2008 line, for MMMs having 30% PPN loads. In all cases, the HPA-MMMs before thermal rearrangement provided good selectivity versus permeability compromise, similar to their thermally rearranged counterparts but in the zone of high selectivity. For H2/CH4, H2/N2, these nonthermally rearranged MMMs approach the 2008 Robeson’s upper bound while H2/CO2 gives selective transport favoring H2 on the 1991 Robeson’s bound. Thus, attending to the energy cost of thermal rearrangement, it could be avoided in some cases especially when high selectivity is the target rather than high permeability.