  We investigate the degradation of quantum entanglement in the
Schwarzschild-de Sitter black hole spacetime, by studying the mutual
information and the logarithmic negativity for maximally entangled, bipartite
states for massless minimal scalar fields. This spacetime is endowed with a
black hole as well as a cosmological event horizon, giving rise to particle
creation at two different temperatures. We consider two independent
descriptions of thermodynamics and particle creation in this background. The
first involves thermal equilibrium of an observer with either of the horizons.
We show that as of the asymptotically flat/anti-de Sitter black holes, in this
case the entanglement or correlation degrades with increasing Hawking
temperatures. The second treats both the horizons combined in order to define a
total entropy and an effective equilibrium temperature. We present a field
theoretic derivation of this effective temperature and show that unlike the
usual cases, the particle creation does not occur here in causally disconnected
spacetime wedges but instead in a single region. Using the associated vacua, we
show that in this scenario the entanglement never degrades but increases with
increasing black hole temperature and holds true no matter how hot the black
hole becomes or how small the cosmological constant is. We argue that this
phenomenon can have no analogue in the asymptotically flat/anti-de Sitter black
hole spacetimes.
