In multihorizon black hole spacetimes, it is possible that there are degene
rate Cauchy horizons with vanishing surface gravities. We investigate the s
tability of the degenerate Cauchy horizon in black hole spacetimes. Despite
the asymptotic behavior of spacetimes (flat, anti-de Sitter, or de Sitter)
, we find that the Cauchy horizon is stable against the classical perturbat
ions, but unstable quantum mechanically. [S0556-2821(99)04908-5].