Entropy of Quantum Fields for Nonextreme Black Holes in the Extreme Limit

Nonextreme black hole in a cavity within the framework of the canonical or grand canonical ensemble can approach the extreme limit with a finite temperature measured on a boundary located at a finite proper distance from the horizon. In spite of this finite temperature, it is shown that the one-loop contribution $S_{q\text{ }}$of quantum fields to the thermodynamic entropy due to equilibrium Hawking radiation vanishes in the limit under consideration. The same is true for the finite temperature version of the Bertotti-Robinson spacetime into which a classical Reissner-Nordstr\"{o}m black hole turns in the extreme limit. The result $S_{q}=0$ is attributed to the nature of a horizon for the Bertotti-Robinson spacetime.