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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.