Double proton transfers (DPTs) are important for several physical processes, both in molecules and in the condensed phase. While these have been widely studied in biological systems, their study in crystalline environments is rare. In this work, using path integral molecular dynamics simulations, we have studied temperature dependent DPT in molecular crystals of terephthalic acid (TPA). In accordance with experimental reports, we find evidence for a double proton transfer induced order-to-disorder transition that is sensitive to the inclusion of nuclear quantum effects. Our simulations show that in addition to the presence of L and R tautomers of terepthalic acid, there are a small but non-negligible concentration of positive and negatively charged pairs of TPA molecules. At the onset of the transition at low temperatures, DPT likely occurs through a tunneling mechanism while at room temperature, likely involving the dominance of activated hopping. Through an analysis of the electronic structure of the system using Wannier functions, we show that the H atom shuttling between the donor and acceptor O atoms involves a proton.