We study the effect of uncorrelated random disorder on the temperature dependence of the superfluid stiffness in the two-dimensional classical XY model. By means of a perturbative expansion in the disorder potential, equivalent to the T-matrix approximation, we provide an extension of the effective-medium-theory result able to describe the low-temperature stiffness, and its separate diamagnetic and paramagnetic contributions. These analytical results provide an excellent description of the Monte Carlo simulations for two prototype examples of uncorrelated disorder. Our findings offer an interesting perspective on the effects of quenched disorder on longitudinal phase fluctuations in two-dimensional superfluid systems.
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