  Pseudospectral analyses have broadened our understanding of ringdown
waveforms from binary remnants, by providing insight into both the stability of
their characteristic frequencies under environmental perturbations, as well as
the underlying transient and non-modal phenomenology that a mode analysis may
miss. In this work we present the pseudospectrum of scalar perturbations on
spherically-symmetric black holes in de Sitter spacetimes. We expand upon
previous analyses in this setting by calculating the pseudospectrum of
Reissner-Nordstr\"om-de Sitter black holes, and revisit results regarding the
stability of quasinormal modes under perturbations in several cases. Of
particular note is the case of scalar quasinormal modes with angular parameter
$\ell=0$, which possess a zero mode related to the presence of a cosmological
horizon. We show that the non-trivial eigenfunction associated to this mode has
a vanishing energy norm which poses a challenge in quantifying the magnitude of
external perturbations to the wave equation's potential, as well as in
calculating the pseudospectrum. Nonetheless, we present results which suggest
that the spectral instability manifestation of $\ell=0$ scalar quasinormal
modes is qualitatively the same as in other cases, in contrast to recent
claims. We also analyze the stability of the fundamental mode for $\ell\ge1$,
finding it to be spectrally stable, except for certain configurations in which
a perturbation leads to a discontinuous overtaking of the fundamental
unperturbed purely-imaginary mode by a perturbed complex quasinormal mode.
