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The effect of Lifshitz transition on thermodynamics and superconductivity in hole-doped cuprates has been heavily debated but remains an unanswered question. In particular, an observed maximum of electronic specific heat is proposed to originate from a quantum critical point associated with the Lifshitz transition. Here, we report an in situ angle-resolved photoemission spectroscopy study of three-dimensional Fermi surfaces in La_2-xSr_xCuO₄ thin films (x = 0.06 – 0.35). With accurate k_z dispersion quantification, the Lifshitz transition is determined to happen across a finite range x = 0.20 – 0.21. Normal state electronic specific heat, calculated from spectroscopy-derived band parameters, agrees with previous thermodynamic microcalorimetry measurements within experimental error. A d-wave superconducting gap smoothly across the Lifshitz transition demonstrates the insensitivity of superconductivity to the dramatic density of states enhancement.