  Gravitational physics is arguably better understood in the presence of a
negative cosmological constant than a positive one, yet there exist strong
technical similarities between the two settings. These similarities can be
exploited to enhance our understanding of the more speculative realm of quantum
cosmology, building on robust results regarding anti-de Sitter black holes
describing the thermodynamics of holographic quantum field theories. To this
end, we study 4-dimensional gravitational path integrals in the presence of a
negative cosmological constant, and with minisuperspace metrics. We put a
special emphasis on boundary conditions and integration contours. The
Hawking-Page transition is recovered and we find that below the minimum
temperature required for the existence of black holes the corresponding saddle
points become complex. When the asymptotic anti-de Sitter space is cut off at a
finite distance, additional saddle points contribute to the partition function,
albeit in a very suppressed manner. These findings have direct consequences for
the no-boundary proposal in cosmology, because the anti-de Sitter calculation
can be brought into one-to-one correspondence with a path integral for de
Sitter space with Neumann conditions imposed at the nucleation of the universe.
Our results lend support to recent implementations of the no-boundary proposal
focusing on momentum conditions at the "big bang".

High Energy Physics - Theory;General Relativity and Quantum Cosmology