The article presents the subsequent development of the established thermohydrogravidynamic theory (intended for deterministic prediction of the temporal intensifications of the global and regional seismotectonic, volcanic and climatic activity of the Earth) based on the author's generalized differential formulation of the first law of thermodynamics extending the classical Gibbs' formulation by taking into account (along with the classical infinitesimal change of heat Q and the classical infinitesimal change of the internal thermal energy dU ) the established differential energy gravitational influence dG (due to the cosmic and terrestrial non-stationary gravitation) on the continuum region along with the established differential increment dK of the macroscopic kinetic energy, the established differential increment d of the gravitational potential energy and the established generalized expression for the differential work Anp, done by the non-potential terrestrial stress forces (determined by the symmetric stress tensor T) acting on the boundary surface of the individual finite continuum region subjected to the non-stationary Newtonian gravitation. Taking into account the combined solar and planetary integral energy gravitational influence of the internal rigid core c,r of the Earth 3 during the considered range (2004 ÷ 2026) AD, the author presents (on May 30, 2018) the foundation (based on the established and confirmed global prediction thermohydrogravidynamic principles determining the strongest temporal intensifications of the global and regional seismotectonic, volcanic and climatic activity of the Earth) of the established first forthcoming subrange (2020 ÷ 2026) AD of the increased intensification of the global seismotectonic, volcanic, climatic and magnetic activity of the Earth related with the maximal (near t * (c,r, 2021)=2021.1 AD)) and the minimal (near t * (c,r, 2021)=2021.65 AD)) combined cosmic (planetary and solar) integral energy gravitational influences on the internal rigid core of the Earth.