A thermodynamically consistent description of induced fission pathways in the ²⁹⁶Lv superheavy nucleus is presented within the framework of nuclear energy-density functional theory. Using self-consistent finite-temperature Hartree-Fock-Bogoliubov calculations with the Skyrme energy-density functional SkM*, the authors derive and compare effective potentials corresponding to different thermodynamic processes isothermal (T=const), isentropic (S=const), and isoenergetic (E=const) as functions of quadrupole deformation and excitation energy. It is shown that at the same amount of excitation energy, the isoenergetic description predicts the highest fission barrier and the largest damping factor. The suppression of the isoenergetic fission barrier is analysed by examining how the driving force changes with increasing excitation energy and by studying the contribution of the nonpotential term to the effective potential. Within the framework of the three thermodynamic schemes considered, the different behaviour of temperature and entropy along the fission pathway is emphasised as well. A transition of the effective level-density parameter from a deformation-sensitive quantity at low excitation energies to a nearly constant value at high energies is observed, in line with the Fermi-gas model expectations.