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Various concepts related to parity-time symmetry, including anti–parity-time symmetry, have found broad applications in wave physics. Wave systems are fundamentally described by Hermitian operators, whereas their unusual properties are introduced by incorporation of gain and loss. We propose that the related physics need not be restricted to wave dynamics, and we consider systems described by diffusive dynamics. We study the heat transfer in two countermoving media and show that this system exhibits anti–parity-time symmetry. The spontaneous symmetry breaking results in a phase transition from motionless temperature profiles, despite the mechanical motion of the background, to moving temperature profiles. Our results extend the concepts of parity-time symmetry beyond wave physics and may offer opportunities to manipulate heat and mass transport.