Following the conclusions of an information theory analysis that hydrophobic hydration is dictated by the equation of state of liquid water, we perform simulations of ten different water models to examine the correlation between the fidelity of each model to the experimental density of liquid water and the accuracy of its description of methane hydration. We find that the three- and five-point water models provide an inferior description of both the liquid density and methane solubility compared to the four-point water models. Of the four-point water models, TIP4P/2005 provides the best description of both the aqueous equation-of-state and methane hydration thermodynamics. When the optimized potentials for liquid simulation united-atom description for methane is used, we find that while the entropy and heat capacity of methane hydration are in excellent agreement with experiment, the chemical potential and enthalpy are systematically shifted upwards. We subsequently reoptimize the methane interaction to accurately reproduce the experimental solubilities as a function of temperature by accounting for missing attractive interactions.