An exciting debate arises when microscopic mechanisms involved in the denaturation of proteins at high pressures are explained. In particular, the issue emerges when the hydrophobic effect is invoked, given that hydrophobicity cannot elucidate by itself the volume changes measured during protein unfolding. In this work, we study by the use of molecular dynamics simulations and essential dynamics analysis the relation between the solvation dynamics, volume, and water structure when apomyoglobin is subjected to a hydrostatic pressure regime. Accordingly, the mechanism of cold denaturation of proteins under high-pressure can be related to the disruption of the hydrogen-bond network of water favoring the coexistence of two states, low-density and high-density water, which directly implies in the formation of a molten globule once the threshold of 200 MPa has been overcome.