Creation of thermal distribution of particles by a black hole is independent of the detail of gravitational collapse, making the construction of the eternal horizons suffice to address the problem in asymptotically flat spacetimes. For eternal de Sitter black holes however, earlier studies have shown the existence of both thermal and non-thermal particle creation, originating from the non-trivial causal structure of these spacetimes. Keeping this in mind we consider this problem in the context of a quasistationary gravitational collapse occurring in a $(3+1)$-dimensional eternal de Sitter, settling down to a Schwarzschild- or Kerr-de Sitter spacetime and consider a massless minimally coupled scalar field. There is a unique choice of physically meaningful `in vacuum here, defined with respect to the positive frequency cosmological Kruskal modes localised on the past cosmological horizon ${cal C^-}$, at the onset of the collapse. We define our `out vacuum at a fixed radial coordinate `close to the future cosmological horizon, ${cal C^+}$, with respect to positive frequency outgoing modes written in terms of the ordinary retarded null coordinate, $u$. We trace such modes back to ${cal C^-}$ along past directed null geodesics through the collapsing body. Some part of the wave will be reflected back without entering it due to the greybody effect. We show that these two kind of traced back modes yield the two temperature spectra and fluxes subject to the aforementioned `in vacuum. Since the coordinate $u$ used in the `out modes is not well defined on a horizon, estimate on how `close we might be to ${cal C^+}$ is given by estimating backreaction. We argue no other reasonable choice of the `out vacuum would give rise to any thermal spectra. Our conclusions remain valid for all non-Nariai class black holes, irrespective of the relative sizes of the two horizons.
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