The generating functional approach to Green's functions in the thermal equilibrium is applied to explore the geometrical origin of the temperatures of the quantum fields in the Rindler spacetime and black hole spacetimes. It is shown that under the transformation from Minkowski space to the Rindler space the path integral representation for the Euclidean generating functionals of Green's functions at zero temperature would transform into the corresponding ones of the quantum fields at a certain finite temperature, and the Minkowski vacuum state would have the same properties as that of the quantum mixed state at that temperature. All thermal Green's functions for the mixed state are given. Similar results would be obtained for the Schwarzschild, the Reissner-Nordstrom and the Kerr black holes and whereupon the Hawking temperature for the black holes would have geometrical origin as well as that in the Eindler spacetime. The various density operators of the mixed states at the Hawking temperature for the black hole spacetimes are specified.