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We discuss the coupled variations of the gravitational, strong andelectroweak coupling constants and the current knowledge of the nuclearequation of state based on heavy ion collision experiments and neutron starmass-radius relationship. In particular we focus in our description onphenomenological parameters, $R$, relating variations in the quantumchromodynamics scale $\Lambda_{QCD}$ and the fine structure constant $\alpha$,and $S$, relating variation of $v$, the Higgs vacuum expectation value and theYukawa couplings, $h$, in the quark sector. This parametrization is valid forany model where gauge coupling unification occurs at some (unspecified) highenergy scale. From a physically motivated set of equations of state for densematter we obtain the constrained parameter phase space $(R, S)$ in high densitynuclear environments. This procedure is complementary to (although currentlyless powerful than) those used in low density conditions. For variations of$\Delta \alpha/\alpha=0.005$ we find that the obtained constrained parameterlies on a strip region in the $(R, S)$ plane that partially overlaps some ofthe allowed values of parameters derived from primordial abundances. This maybe of interest in the context of unification scenarios where a dense phase ofthe universe may have existed at early times.

Constraints of the variation of fundamental couplings and sensitivity of the equation of state of dense matter