A secure key distribution (exchange) scheme is unconditionally secure if it is unbreakable against arbitrary technological improvements of computing power and/or any development of new algorithms. There are only two families of experimentally realized and tested unconditionally secure key distribution technologies: quantum key distribution (QKD), the base of quantum cryptography, which utilizes quantum physical photonic features, and the Kirchhoff-Law–Johnson-Noise (KLJN) system that is based on classical statistical physics (fluctuation–dissipation theorem). The focus topic of this paper is the thermodynamical situation of the KLJN system. In all the original works, the proposed KLJN schemes required thermal equilibrium between the devices of the communicating parties to achieve perfect security. However, Vadai et al., in (Nature) Sci. Rep. 5, 13653 (2015) show a modified scheme, where there is a non-zero thermal noise energy flow between the parties, yet the system seems to resist all the known attack types. We introduce an attack type against their system. The attack utilizes coincidence events between the line current and voltages. We show that there is a non-zero information leak toward the Eavesdropper, even under idealized conditions. As soon as the thermal equilibrium is restored, the system becomes perfectly secure again. In conclusion, perfect unconditional security requires thermal equilibrium.