  A local Hawking temperature is derived for any future outer trapping horizon
in spherical symmetry, using a Hamilton-Jacobi variant of the Parikh-Wilczek
tunneling method. It is given by a dynamical surface gravity as defined
geometrically. The operational meaning of the temperature is that Kodama
observers just outside the horizon measure an invariantly redshifted
temperature, diverging at the horizon itself. In static, asymptotically flat
cases, the Hawking temperature as usually defined by the Killing vector agrees
in standard cases, but generally differs by a relative redshift factor between
the horizon and infinity, being the temperature measured by static observers at
infinity. Likewise, the geometrical surface gravity reduces to the Newtonian
surface gravity in the Newtonian limit, while the Killing definition instead
reflects measurements at infinity. This may resolve a longstanding puzzle
concerning the Hawking temperature for the extremal limit of the charged
stringy black hole, namely that it is the local temperature which vanishes. In
general, this confirms the quasi-stationary picture of black-hole evaporation
in early stages. However, the geometrical surface gravity is generally not the
surface gravity of a static black hole with the same parameters.
