  The thermodynamic instability, for example the negative heat capacity, of a
black hole implies the existence of off-shell negative mode(s) (tachyonic
mode(s)) around the black hole geometry in the Euclidean path integral
formalism of quantum gravity. We explicitly construct an off-shell negative
mode inspired from the negative heat capacity in the case of Schwarzschild
black hole with/without a cosmological constant. We carefully check the
boundary conditions, i.e. the regularity at the horizon, the traceless
condition, and the normalizability.
