  Using the 3rd quantization formalism we study the quantum entanglement of
universes created in pairs within the framework of standard homogeneous and
isotropic cosmology. In particular, we investigate entanglement quantities
(entropy, temperature) around maxima, minima and inflection points of the
classical evolution. The novelty from previous works is that we show how the
entanglement changes in an extended minisuperspace parameterised by the scale
factor and additionally, by the massless scalar field. We study the
entanglement quantities for the universes which classically exhibit Big-Bang
and other than Big-Bang (exotic) singularities such as Big-Brake, Big-Freeze,
Big-Separation, and Little-Rip. While taking into account the scalar field, we
find that the entanglement entropy is finite at the Big-Bang singularity and
diverges at maxima or minima of expansion. As for the exotic singularity models
we find that the entanglement entropy or the temperature in all the critical
points and singularities is either finite or infinite, but it never vanishes.
This shows that each of the universes of a pair is entangled to a degree
parametrized by the entanglement quantities which measure the quantumness of
the system. Apart from the von Neumann entanglement entropy, we also check the
behaviour of the the Tsallis and the Renyi entanglement entropies, and find
that they behave similarly as the meters of the quantumness. Finally, we find
that the best-fit relation between the entanglement entropy and the Hubble
parameter (which classically marks special points of the universe evolution) is
of the logarithmic shape, and not polynomial, as one could initially expect.
