We study the thermodynamical properties of compressed baryonic matter with strangeness within non-relativistic energy density functional models with a particular emphasis on possible phase transitions found earlier for a simple $n,p,e,Lambda$-mixture. The aim of the paper is twofold: I) examining the phase structure of the complete system, including the full baryonic octet and II) testing the sensitivity of the results to the model parameters. We find that, associated to the onset of the different hyperonic families, up to three separate strangeness-driven phase transitions may occur. Consequently, a large fraction of the baryonic density domain is covered by phase coexistence with potential relevance for (proto)-neutron star evolution. It is shown that the presence of a phase transition is compatible both with the observational constraint on the maximal neutron star mass, and with the present experimental information on hypernuclei. In particular we show that two solar mass neutron stars are compatible with important hyperon content. Still, the parameter space is too large to give a definitive conclusion of the possible occurrence of a strangeness driven phase transition, and further constraints from multiple-hyperon nuclei and/or hyperon diffusion data are needed.
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