Experiments have pointed to the formation of the electron quadrupling condensate in Ba_1−xK_xFe₂As₂ at x ~ 0.8. The state spontaneously breaks time-reversal symmetry and is sandwiched between two critical points, separating it from the broken time-reversal symmetry (BTRS) superconducting state at TcU(1) and a normal-metal state at TcZ2. We report a theory of the acoustic effects spectroscopy of systems with an electron quadrupling phase based on ultrasound-velocity measurements. We show that the experimental results are consistent with BTRS superconductivity at x ~ 0.8, fulfilling the necessary condition for the formation of electron quadrupling in Ba_1−xK_xFe₂As₂. We provide the theoretical basis and the experimental strategy to study the order parameter symmetry of emerging quadrupling condensates in superconductors.