Abstract
It is shown that in the extreme limit with a zero surface gravity but nonzero local temperature the limiting metric of a generic static black hole is determined by a metric induced on a horizon and one function of two coordinates, stress-energy tensor of a source picking up its values from a horizon. The limiting procedure is extended to rotating black holes. If the extreme limit is due to merging a black hole horizon and cosmological one both horizons are always in thermal equilibrium in this limit. This is proved for a generic case of static or axially-symmetrical rotating spacetimes.
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