Hawking effect of AdS$_2$ black hole in the Jackiw-Teitelboim model

It might be tempting to consider that the two-dimensional anti-de Sitter black hole in the Jackiw-Teitelboim model is thermally hot by invoking the non-vanishing surface gravity. So, one might expect that the local temperature would also be blue-shifted near the horizon, while it would vanish at infinity because of the Tolman factor of the local temperature. In this paper, we will show that the local temperature vanishes everywhere, which respects the equivalence principle perfectly in contrast to the case of the asymptotically flat black holes recovering the equivalence principle just at the horizon. In other words, the origin of radiation for a fixed observer comes from the Unruh effect rather than the Hawking effect. If the inward and outward fluxes are defined non-trivially at the horizon, then the local temperature of the black hole takes the usual Tolman form; however, the divergent Tolman temperature at the horizon is incompatible with the finite local energy density there.