Local temperature for dynamical black holes
  A local Hawking temperature was recently derived for any future outer trapping horizon in spherical symmetry, using a Hamilton-Jacobi tunneling method, and is given by a dynamical surface gravity as defined geometrically. Descriptions are given of the operational meaning of the temperature, in terms of what observers measure, and its relation to the usual Hawking temperature for static black holes. Implications for the final fate of an evaporating black hole are discussed. 
Sean A. Hayward, R. Di Criscienzo, M. Nadalini, L. Vanzo, S. Zerbini