The lack of thermodynamic consistency is a well-recognized problem in the single-component pseudopotential lattice Boltzmann models which prevents them from replicating accurate liquid and vapor phase densities; i.e., current models remain unable to exactly match coexisting density values predicted by the associated thermodynamic model. Most of the previous efforts had attempted to solve this problem by introducing tuning parameters, whose determination required empirical trial and error until acceptable thermodynamic consistency was achieved. In this study, we show that the problem can be alternatively solved by properly designing customized equations of state (EOSs) that replace any cubic EOS of choice during the computation of effective mass used in Shan-Chen forces. A two-parameter cubic-shaped customized EOS is introduced. Contrary to previous efforts, customization parameters in the new EOS are nonempirical and are rather derived from solving the integral mechanical stability equation, which neglects the need for any type of tuning for the attainment of rigorous thermodynamic consistency. The proposed approach reduces the errors of the coexisting densities and saturated pressure in the simulation to a maximum of 0.01% within the liquid-vapor density ratio range from O(1) to O(10^{4}), which had not been achieved in any of the previous tuning-based efforts. A straightforward way for achieving the desired surface tension via the customized EOS is also provided.