The relationship between the threshold energy for a deep sub-barrier fusion hindrance phenomenon and the energy at which the regime of interaction changes (the turning-off of the nuclear forces and friction) in the sub-barrier capture process, is studied within the quantum diffusion approach. The quasielastic barrier distribution is shown to be a useful tool to clarify whether the slope of capture cross section changes at sub-barrier energies.The experiments with various medium-light and heavy nuclei have shown that the experimental slopes of the complete fusion excitation function keep increasing at low sub-barrier energies and may become much larger than the predictions of standard coupled-channel calculations. This was identified as the fusion hindrance with the threshold energy E s [1-3]. More experimental and theoretical studies of sub-barrier fusion hindrance are required to improve our understanding of its physical reason, which may be especially important in astrophysical fusion reactions [4].As shown within the quantum diffusion approach [5-9], due to a change of the regime of interaction (the turningoff of the nuclear forces and friction) at deep sub-barrier energies, the curve related to the capture cross section as a function of bombarding energy has smaller slope. In the present paper we try to demonstrate the relationship between the threshold energy E s for a deep sub-barrier fusion hindrance phenomenon and the energy E ch at which the regime of interaction changes in the sub-barrier capture process.In the quantum diffusion approach the capture of nuclei is treated in terms of a single collective variable: the relative distance between the colliding nuclei. The neutron transfer and nuclear deformation effects are taken into consideration through the dependence of the nucleus-nucleus potential on the isotopic compositions, deformations and orientations of interacting nuclei. Our approach takes into consideration the fluctuation and dissipation effects in collisions of heavy ions which model the coupling with various channels (for example, coupling of the relative motion with low-lying collective modes such as dynamical quadrupole and octupole modes of target and projectile [10]). We have to mention that many quantum-mechanical and non-Markovian effects [11][12][13] accompanying the passage through the potential barrier are taken into consideration in our formalism [5][6][7][8][9]. The details of used formalism are presented in our previous articles [5,6]. With this approach many heavy-ion capture reactions at energies above and well below the Coulomb barrier have been successfully described.Within the quantum diffusion model [5-9] the nuclear forces start to play a role at relative distance R int = R b + 1.1 fm (R b is the position of the Coulomb barrier at given angular momentum and orientations of the interacting nuclei) where the nucleon density of colliding nuclei approximately reaches 10% of saturation density. If the colliding nuclei approach the distance R int between their centers, the nucle...