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Interest in cosmological singularities has remarkably grown in recent times,particularly on future singularities with the discovery of late-timeacceleration of the universe and dark energy. While such explorations havepreviously been done in various modified gravity and quantum gravitationallyinspired cosmologies (besides standard general relativistic cosmology), no suchan endeavour has been taken up till now in the realms of renormalization groupapproaches to cosmology and we have hence took up on this journey. In thiswork, we consider the formation of cosmological singularities in anasymptotically safe cosmology where the cut off scale is proportional to theHubble parameter. We consider a well motivated inhomogeneous form of theequation of state(EOS) as well. We firstly delve into some basics of thiscosmology and show that such a scenario permits a transition between phantomand quintessence forms of universal evolution. We then show that one can haveType I - Type IV singularities in such a cosmology for various version of theEOS and that the singularities can occur both in finite and infinite time. Theconditions in which these singularities occur is significantly different thanhow they occur in the standard cosmology, with the formulations being even moreinvolved. Interestingly this difference comes about without the need of anyfree parameters in the cosmological models, which is usually what one observeswhen one goes for the same pursuit via various modified gravity/ QG inspiredcosmological approaches. Furthermore, we showed that usual singularity removalmethods like conformal anomaly effects and f(R) gravity effects do not offermuch hope for singularity removal in this cosmology.

Singularity formation in asymptotically safe cosmology with inhomogeneous equation of state