We show that the cusp-like structure in the nuclear symmetry energy, a consequence of topology change, visible in the skyrmion lattice description of dense matter is extremely robust. It is present in the Skyrme model -with the pion field only -and is left unscathed by massive degrees of freedom such as the vector mesons ρ and ω and also the scalar meson introduced as a dilaton. It leads to the emergence of parity-doubling at the skyrmion-half-skyrmion transition and impacts on the nuclear symmetry energy "wilderness" landscape, weeding out roughly half of the wilderness. It is shown that a pseudo-conformal sound velocity arises at a precocious density at which the cusp is formed. It is suggested that the topology change, being robust, could impact on the tidal deformability observed in gravity waves PACS numbers: 12.39.Dc 12.39.Fe 21.65.Ef