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Experimental in-situ studies have shown that the compressional behavior of zeolite frameworks depends not only on the framework topology, but also on the extra-framework content, i.e. the species residing in the pores (cations, water molecules, templates). However, systematic experimental studies of isotypic frameworks with different extra-framework content often face several challenges, such as limited quality of the diffraction data and non-ideal behavior of the pressure-transmitting medium. In order to provide an alternative computational perspective, the compressional behavior of four fluoroaluminophosphate zeolite analogues with a chabazite (CHA) topology was investigated using electronic structure calculations in the framework of dispersion-corrected density functional theory (DFT). The AlPO-CHA systems differ in the nature of the organic template molecule occupying the chabazite cage, with the template species being morpholinium (morph), pyridinium (pyr), dimethylimidazolium (DMI), and trimethylimidazolium (TMI), respectively. For the case of AlPO-CHA_morph, the evolution of the lattice parameters with pressure as obtained from the computations was found to be in agreement with experimental data from an earlier in-situ diffraction investigation. In particular, the occurrence of a structural transition at pressures above 3 GPa could be reproduced. An analysis of the DFT-optimized structures showed that a pronounced elongation of the chabazite cage occurs upon the transition, which is associated with a rearrangement of the morpholinium molecules residing in the cage. Moreover, the double six-ring building units are strongly deformed during the transition. Analogous computations for the other three AlPO-CHA systems showed no indications for the occurrence of a P-induced transition, with only minor discontinuities in the evolution of the structural parameters with pressure. The compressibility in the low-pressure range varied widely depending on the template, with bulk moduli ranging from 19 GPa for AlPO-CHA_pyr to 51 GPA for AlPO-CHA_TMI. Despite the identical composition of the framework, different deformations of the double six-ring units with pressure were observed. The present study highlights the important influence of the template on the compressional behavior of porous framework compounds. Firstly, the molecular volume of the template and the nature of the framework-template interactions impact upon the compressibility in the low-pressure regime. Secondly, the ability of the template to accommodate pressure through continuous rotation and shortening of template-template and framework-template distances determines whether a discontinuous structural transition occurs or not.