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

For the higher-dimensional topological dS spacetime with the non-linear source, there are two horizons with different radiation temperatures. We can regard this system as an ordinary thermodynamic system in the thermodynamic equilibrium, which is of the thermaldynamic parameters ($T_{eff},~P_{eff},~V,~S,~\Phi_{eff}$). In this work, our focus is on the thermodynamic properties of the phase transition for the four-dimensional dS spacetime with different charge corrections $\bar\phi$. The critical thermadynamic parameters are exhibited. We find there exists phase transition in diagrames of $P_{eff}-V$ and $G_{T_{eff}}-P_{eff}$ nearby the critical point. The isocratic heat capacity $C_{P_{eff}}$ has the schottky peak at the critical point. These results are similar to that of AdS black hole. However, it is unique the isobaric heat capacity $C_{V_{eff}}$ for the dS spacetime is not zero. Furthermore, we also analyze the effect of the non-linear source on the properies of the phase transtion for this system.