Revisiting black hole thermodynamics in massive gravity: charged particle absorption and shell of dust falling

We study the variation of the massive black hole in both normal and extended thermodynamic phase spaces via two methods. The first method includes a charged particle absorbed by the black hole, while the second method includes a shell of dust falling into the black hole. Using the former method, the first and second laws of thermodynamics are always satisfied in the normal phase space, while in the extended phase space, the first law of thermodynamics is also satisfied, but the validity of the second law of thermodynamics depends on the model parameters. Applying the latter method, the first and second laws of thermodynamics are both valid. We argue that the possible reason for the violation of the second law of thermodynamics via the former method may be attributed to the assumption that the change of internal energy of the black hole is equal to the energy of the particle. Finally, we show that the event horizon always exists to guarantee the validity of the weak cosmic censorship in both phase spaces, which means that the violation of the second law of thermodynamics under the assumption does not affect the weak cosmic censorship conjecture. This further supports our argument that the assumption in the first method may be the reason for the violation of the law and it requires deeper treatment.