  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 $\rm
La_{2-x}Sr_xCuO_4$, whose superconductivity features differ from what is
described by the classical Bardeen-Cooper-Schrieffer theory [J.I. Bo\^zovi\'c,
X. He, J. Wu, and A. T. Bollinger, Nature 536, 309 (2016)]. We demonstrate that
1) at temperature $T=0$, the superfluid density $n_s$ turns out to be
considerably smaller than the total electron density; 2) the critical
temperature $T_c$ is controlled by $n_s$ rather than by doping, and is a linear
function of the $n_s$; 3) at $T>T_c$ the resistivity $\rho(T)$ varies linearly
with temperature, $\rho(T)\propto \alpha T$, where $\alpha$ diminishes with
$T_c\to 0$, while in the normal overdoped (non superconducting) region with
$T_c=0$, the resistivity becomes $\rho(T)\propto T^2$. 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.
