We show that a topological quantum phase transition, generating flat bands and altering Fermi surface topology, is a primary reason for the exotic behavior of the overdoped high-temperature superconductors represented by La2−xSrxCuO4, whose superconductivity features differ from what is described by the classical Bardeen-Cooper-Schrieffer theory [J.I. Boẑović, X. He, J. Wu, and A. T. Bollinger, Nature 536, 309 (2016)]. We demonstrate that 1) at temperature T = 0, the superfluid density ns turns out to be considerably smaller than the total electron density; 2) the critical temperature Tc is controlled by ns rather than by doping, and is a linear function of the ns; 3) at T > Tc the resistivity ρ(T ) varies linearly with temperature, ρ(T ) ∝ αT , where α diminishes with Tc → 0, while in the normal overdoped (non superconducting) region with Tc = 0, the resistivity becomes ρ(T ) ∝ T 2 . The theoretical results presented are in good agreement with recent experimental observations, closing the colossal gap between these empirical findings and Bardeen-Cooper-Schrieffer-like theories.