The double Hawking temperature T=2T(H) appears in some approaches to the Hawking radiation when the radiation is considered in terms of the quantum tunneling. We consider the origin of such unusual temperature for the black hole horizon and also for the cosmological horizon in de Sitter spacetime. In the case of the black hole horizon, there are two contributions to the tunneling process of radiation, each being governed by the temperature T=2T(H). These processes are coherently combined to produce the radiation with the Hawking temperature T-H. This can be traditionally interpreted as the pair creation of two entangled particles, of which one goes towards the center of the black hole, while the other one escapes from the black hole. In the case of the cosmological horizon, the temperature T=2T(H) is physical. While the creation of the entangled pair is described by the Hawking temperature, the de Sitter spacetime allows for another process, in which only a single (non-entangled) particle inside the cosmological horizon is created. This process is characterized by the local temperature T=2T(H). The local single-particle process also takes place outside the black hole horizon, but it is exponentially suppressed.