Can Hawking temperatures be negative ?

It has been widely believed that the Hawking temperature for a black hole is uniquely determined by its metric and positive. But I find that this is ``not'' true in the recently discovered black holes which include the exotic black holes and the black holes in the three-dimensional higher curvature gravities. I show that the Hawking temperatures, which are measured by the quantum fields in thermal equilibrium with the black holes, are not the usual Hawking temperature but the new temperatures that have been proposed recently and can be negative. The associated new entropy formulae, which are defined by the first law of thermodynamics, versus the black hole masses show some genuine effects of the black holes which do not occur in the spin systems. Some cosmological implications are noted also