Continuous Phase Transition of the higher-dimensional topological de-Sitter Spacetime with the Non-linear Source

For the higher-dimensional dS spacetime embedded with black holes with non-linear charges, there are two horizons with different radiation temperatures. By introducing the interplay between two horizons this system can be regarded as an ordinary thermodynamic system in the thermodynamic equilibrium described by the thermodynamic quantities ($T_{eff},~P_{eff},~V,~S,~\Phi_{eff}$). In this work, our focus is on the thermodynamic properties of phase transition for the four-dimensional dS spacetime with different values of the charge correction $\bar\phi$. We find that with the increasing of the non-linear charge correction the two horizons get closer and closer, and the correction entropy is negative which indicates the interaction between the two horizons stronger and stronger. Furthermore, the heat capacity at constant pressure, isobaric expansion coefficient, and the isothermal compression coefficient have the schottky peak at the critical point. However, the heat capacity as constant volume for the dS spacetime is nonzero. Finally, the dynamical properties of phase transition for this system have investigated based on Gibbs free energy, where exists the different behavior with that for AdS black holes.