Warm-mix asphalt technology has been applied to recycled rubber asphalt binder (RAB), which forms warm-mixed crumb rubber-modified asphalt binder (W-RAB) as a “green” material for environmental conservation and to enhance road performance. Furthermore, low-temperature cracking is one of the major distresses for asphalt pavement, which drastically restricts ride quality and service level. Therefore, the main objective of this study is to comparatively analyze the low-temperature properties of W-RABs based on thermal stress and the simple fractional model. W-RABs were obtained by mixing 60 mesh recycled rubber (CR) and two different types of warm-mix additives, namely viscosity reducer (1, 2, and 3%) and surfactant (0.4, 0.6, and 0.8%). First, Hopkins and Hamming’s numerical algorithm and the Boltzmann superposition principle were used for obtaining thermal stress σ(T). Subsequently, critical cracking temperature Tcr was derived using the single asymptote procedure (SAP) theory. Second, the simple fractional viscoelasticity model was used to calculate the creep compliance, damping ratio, and dissipation energy ratio, and the results were compared with the Superpave protocol results obtained with bending beam rheometer (BBR) tests. The results showed that a combination of CR and warm-mix additives could slightly improve the thermal crack resistance of the asphalt binder. The addition of 0.6% surfactant yielded the optimum performance, while only a high dosage (3%) of viscosity reducer provided a marked improvement in efficiency, which decreased with a decrease in temperature. This study recommends the use of RAB composited with 0.6% surfactant for areas with extremely low temperature.