This study examines the impact of the recycled brick powder (RBP) replacement rate, especially at elevated temperatures on RBP-ultra-high-performance concrete (UHPC) properties such as the stress–strain curve, Poisson’s ratio, elastic modulus, and axial compressive strength through uniaxial compression experiments. The results show that with the increase of heating temperature, the axial compressive strength of the specimen increases first and then decreases under natural cooling (NC). In contrast, Poisson’s ratio shows opposite values. The peak strain continues to increase, and the initial elastic modulus and peak secant modulus continue to decrease. Compared with NC, the axial compressive strength of the specimens under water cooling has been reduced, the peak strain is generally larger, the initial elastic modulus and the peak secant modulus are smaller, and the incorporation of RBP also has a certain effect on the mechanical properties. Through regression analysis, an equation is established to calculate the axial compressive strength of RBP-UHPC with temperature, accounting for variables such as temperature, RBP replacement rate, and cooling method. Furthermore, based on the results of axial compression experiments, a constitutive equation for axial compression in RBP-UHPC after exposure to high temperatures is proposed. Overall, the theoretical curve closely aligns with the experimental curve, verifying its accuracy.