We examine in detail the recent proposal that extreme cosmic ray dominated regions (CRDRs) characterize the interstellar medium of galaxies during events of high‐density star formation, fundamentally altering its initial conditions (Papadopoulos 2010). Solving the coupled chemical and thermal state equations for dense UV‐shielded gas reveals that the large CR energy densities in such systems [UCR ∼ few × (103–104) UCR, Gal] will indeed raise the minimum temperature of this phase (where the initial conditions of star formation are set) from ∼10 K (as in the Milky Way) to ∼50–100 K. Moreover in such extreme CRDRs the gas temperature remains fully decoupled from that of the dust, with Tkin≫Tdust, even at high densities [n(H2) ∼ 105–106 cm−3], quite unlike CRDRs in the Milky Way where Tk∼Tdust when n(H2) ≳ 105 cm−3. These dramatically different star formation initial conditions will (i) boost the Jeans mass of UV‐shielded gas regions by factors of ∼10–100 with respect to those in quiescent or less extreme star‐forming systems and (ii) ‘erase’ the so‐called inflection point of the effective equation of state of molecular gas. Both these effects occur across the entire density range of typical molecular clouds, and may represent a new paradigm for all high‐density star formation in the Universe, with CRs as the key driving mechanism, operating efficiently even in the high dust extinction environments of compact extreme starbursts. The characteristic mass of young stars will be boosted as a result, naturally yielding a top‐heavy stellar initial mass function (IMF) and a bimodal star formation mode (with the occurrence of extreme CRDRs setting the branching point). Such CRDRs will be present in Ultra‐Luminous Infrared Galaxies (ULIRGs) and merger‐driven gas‐rich starbursts across the Universe where large amounts of molecular gas rapidly dissipate towards compact disc configurations where they fuel intense starbursts. In hierarchical galaxy formation models, CR‐controlled star formation initial conditions lend a physical basis for the currently postulated bimodal IMF in merger/starburst versus quiescent/disc star‐forming environments, while naturally making the integrated galactic IMFs a function of the star formation history of galaxies.