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For an FLRW model, thermodynamic phase transitions are investigated in theEinstein-Gauss-Bonnet gravity framework. Using the work density, the equationof state is derived, and the criticality conditions are employed to determinethe critical points where possible phase transitions occur. The appearance ofphase transitions strongly depends on the space-time dimension $n$. In thisconcern, for $n=5$, there is an ``inverted'' first-order phase transition,where the Gibbs free energy presents a swallow-tail behavior. On the otherhand, for $n=6$, the system does not exhibit first order phase transition. Insuch a case, the Gibbs free energy presents a cusp with stable and unstablebranches. For the present study, the mentioned phenomena are present for anexpanding cosmology, where the matter distribution filling the Universecorresponds to a speculative matter distribution with an equation of stateparameter greater than one. Interestingly, there are no phase transitions fordimensions greater than $n=6$, nor for expanding or contracting cosmologicalscenarios. To gain more insights into the system, the microstructure isanalyzed using thermodynamic geometry to quantify the normalized scalarcurvature. This invariant shows that an attractive interaction dominates thephase-transition region. Additionally, the topological thermodynamic analysiswas performed employing Duan's off-shell map. This study reveals that for $n=5$we observe a winding number interchange twice, indicating an unstablesmall/large branch phase transition through an intermediate stable phase. For$n=6$ the number of exotic defects is one. Consequently, we observe a stablesmall branch and an unstable large branch.

Thermodynamic aspects of FLRW Universe in Einstein-Gauss-Bonnet domains