We consider directed polymer models where a fluctuating path is coupled with a random environment. Our focus is on the models with a random environment given by inhomogeneous parameters. We study the limiting free energy of fairly general inhomogeneous models. First, we derive the existence and basic properties of the limiting free energy for an asymptotically mean stationary model. Second, we apply our results to the exactly solvable log-gamma polymer. We give a variational formula for the point to point free energy in terms of the marginal distributions of the parameters. We identify critical angles at which the free energy transitions from strictly concave to linear. We also obtain explicit formulas for some special distributions of the parameters. Third, we study the fluctuation of free energy around some limit shape. We give scaling exponents for the log-gamma polymer. The KPZ exponent 1/3 appears in the concave sector and the diffusive exponent 1/2 in the flat region.