We focus on the response of mechanically stable (MS) packings of frictionless, bidisperse disks to thermal fluctuations, with the aim of quantifying how nonlinearities affect system properties at finite temperature. In contrast, numerous prior studies characterized the structural and mechanical properties of MS packings of frictionless spherical particles at zero temperature. Packings of disks with purely repulsive contact interactions possess two main types of nonlinearities, one from the form of the interaction potential (e.g., either linear or Hertzian spring interactions) and one from the breaking (or forming) of interparticle contacts. To identify the temperature regime at which the contact-breaking nonlinearities begin to contribute, we first calculated the minimum temperatures T-cb required to break a single contact in the MS packing for both single-and multiple-eigenmode perturbations of the T = 0 MS packing. We find that the temperature required to break a single contact for equal velocity-amplitude perturbations involving all eigenmodes approaches the minimum value obtained for a perturbation in the direction connecting disk pairs with the smallest overlap. We then studied deviations in the constant volume specific heat (C) over barV and deviations of the average disk positions Delta r from their T = 0 values in the temperature regime T-(C) over barV < T < Tr, where Tr is the temperature beyond which the system samples the basin of a new MS packing. We find that the deviation in the specific heat per particle Delta(C) over bar (0)(V)/(C) over bar (0)(V) relative to the zero-temperature value (C) over bar (0)(V) can grow rapidly above Tcb; however, the deviation Delta(C) over bar (0)(V)/(C) over bar (0)(V) decreases as N-1 with increasing system size. To characterize the relative strength of contact-breaking versus form nonlinearities, we measured the ratio of the average position deviations Delta r(ss)/Delta r(ds) for single- and double-sided linear and nonlinear spring interactions. We find that Delta r(ss)/Delta r(ds) > 100 for linear spring interactions is independent of system size. This result emphasizes that contact-breaking nonlinearities are dominant over form nonlinearities in the low-temperature range Tcb < T < Tr for model jammed systems.