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We present simulations of galaxy formation, based on the gadget -3 code, in which a sub-resolution model for star formation and stellar feedback is interfaced with a new model for active galactic nucleus (AGN) feedback. Our sub-resolution model describes a multiphase interstellar medium (ISM), accounting for hot and cold gas within the same resolution element: we exploit this feature to investigate the impact of coupling AGN feedback energy to the different phases of the ISM over cosmic time. Our fiducial model considers that AGN feedback energy coupling is driven by the covering factors of the hot and cold phases. We perform a suite of cosmological hydrodynamical simulations of disc galaxies (⁠|$M_{\rm halo,DM} \simeq 2 \times 10^{12}$|  M⊙, at z  = 0), to investigate: (i) the effect of different ways of coupling AGN feedback energy to the multiphase ISM; (ii) the impact of different prescriptions for gas accretion (i.e. only cold gas, both cold and hot gas, with the additional possibility of limiting gas accretion from cold gas with high angular momentum); (iii) how different models of gas accretion and coupling of AGN feedback energy affect the coevolution of supermassive black holes (BHs) and their host galaxy. We find that at least a share of the AGN feedback energy has to couple with the diffuse gas, in order to avoid an excessive growth of the BH mass. When the BH only accretes cold gas, it experiences a growth that is faster than in the case in which both cold and hot gas are accreted. If the accretion of cold gas with high angular momentum is reduced, the BH mass growth is delayed, the BH mass at z  = 0 is reduced by up to an order of magnitude, and the BH is prevented from accreting below z ≲ 2, when the galaxy disc forms. [ABSTRACT FROM AUTHOR]