jats:titleAbstract</jats:title>jats:pIt has been observed that many spacetimes which feature a near-extremal horizon exhibit the phenomenon of zero-damped modes. This is characterised by the existence of a sequence of quasinormal frequencies which all converge to some purely imaginary number jats:inline-formulajats:alternativesjats:tex-math$$i\alpha $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mii</mml:mi> mml:miα</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> in the extremal limit and cluster in a neighbourhood of the line jats:inline-formulajats:alternativesjats:tex-math$${{,\mathrm{Im},}}s=\alpha $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mrow <mml:mspace /> mml:miIm</mml:mi> <mml:mspace /> </mml:mrow> mml:mis</mml:mi> mml:mo=</mml:mo> mml:miα</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. In this paper, we establish that this property is present for the conformal Klein–Gordon equation on a Reissner–Nordström–de Sitter background. This follows from a similar result that we prove for a class of spherically symmetric black hole spacetimes with a cosmological horizon. We also show that the phenomenon of zero-damped modes is stable to perturbations that arise through adding a potential.</jats:p>