Spin-orbit (SO) coupling leads to numerous phenomena in electron systems. Artificial SO coupling in ultracold neutral atoms provides the opportunity to study such phenomena in bosonic systems, which exhibit superfluidity and various symmetry-breaking condensate phases. In general, a richer structure of symmetry breaking results in a nontrivial finite-temperature phase diagram, but the thermodynamics of the SO-coupled Bose gas at finite temperature remains unknown both in theory and experiment. Here we experimentally determine a new finite-temperature phase transition that is consistent with the transition between the stripe ordered phase and the magnetized phase. We also observe that the magnetic phase and the Bose condensate transitions occur simultaneously as temperature decreases. We determine the entire finite-temperature phase diagram of the SO-coupled Bose gas, thus illustrating the power of quantum simulation.