The background temperature of the protoplanetary disk is a fundamental but poorly constrained parameter that strongly influences a wide range of conditions and processes in the early Solar System, including the widespread process(es) by which chondrules originate. Chondrules, mm-scale objects composed primarily of silicate minerals, were formed in the protoplanetary disk almost entirely during the first four million years of Solar System history but their formation mechanism(s) are poorly understood. Here we present new constraints on the sub-silicate solidus cooling rates of chondrules at <873 K (600°C) using the compositions of sulfide minerals. We show that chondrule cooling rates remained relatively rapid (~100 to 101 K/hr) between 873 and 503 K, which implies a protoplanetary disk background temperature of <503 K (230°C) and is consistent with many models of chondrule formation by shocks in the solar nebula, potentially driven by the formation of Jupiter and/or planetary embryos, as the chondrule formation mechanism. This protoplanetary disk background temperature rules out current sheets and resulting short-circuit instabilities as the chondrule formation mechanism. More detailed modeling of chondrule cooling histories in impacts is required to fully evaluate impacts as a chondrule formation model. These results motivate further theoretical work to understand the expected thermal evolution of chondrules at ≤873 K under a variety of chondrule formation scenarios.