Negative modes of Schwarzschild black hole in Einstein-Gauss-Bonnet Theory

We study non conformal negative modes of Schwarzschild black holes in the five dimensional Einstein-Gauss-Bonnet gravity theory with a zero or negative cosmological constant, in the context of the semiclassical quantum gravity formulated by the path integral. We find an unique negative mode when the black hole has a negative heat capacity which is the same in Einstein theory. On the other hand, we still find negative modes inside a parameter region where a small black hole has a positive heat capacity. The number is one/two for the case of zero/negative cosmological constant respectively. In the rest of the parameter region where the heat capacity is positive, we find no negative modes. We discuss the possible physical understanding of having one or two negative modes.