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 does ``not'' seem to be 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 and physical origin of the discrepancy with the standard analysis are noted also.