Effective Temperatures and Radiation Spectra for a Higher-Dimensional Schwarzschild-de-Sitter Black-Hole

The absence of a true thermodynamical equilibrium for an observer located in the causal area of a Schwarzschild-de Sitter spacetime has repeatedly raised the question of the correct definition of its temperature. In this work, we consider five different temperatures for a higher-dimensional Schwarzschild-de Sitter black hole: the bare $T_0$, the normalised $T_{BH}$ and three effective ones given in terms of both the black hole and cosmological horizon temperatures. We find that these five temperatures exhibit similarities but also significant differences in their behaviour as the number of extra dimensions and the value of the cosmological constant are varied. We then investigate their effect on the energy emission spectra of Hawking radiation. We demonstrate that the radiation spectra for the normalised temperature $T_{BH}$ -- proposed by Bousso and Hawking over twenty years ago -- leads to the dominant emission curve while the other temperatures either support a significant emission rate only at a specific $\Lambda$ regime or they have their emission rates globally suppressed. Finally, we compute the bulk-over-brane emissivity ratio and show that the use of different temperatures may lead to different conclusions regarding the brane or bulk dominance.