  In a previous paper we obtained formulae for the volume of a causal diamond
or Alexandrov open set $I^+(p) \cap I^-(q)$ whose duration $\tau(p,q) $ is
short compared with the curvature scale. In the present paper we obtain
asymptotic formulae valid when the point $q$ recedes to the future boundary
${\cal I}^+$ of an asymptotically de-Sitter spacetime. The volume (at fixed
$\tau$) remains finite in this limit and is given by the universal formula
$V(\tau) = {4\over 3}\pi (2\ln \cosh{\tau\over 2}-\tanh^2{\tau\over 2})$ plus
corrections (given by a series in $e^{-t_q}$) which begin at order $e^{-4t_q}$.
The coefficents of the corrections depend on the geometry of ${\cal I}^+$. This
behaviour is shown to be consistent with the no-hair property of cosmological
event horizons and with calculations of de-Sitter quasinormal modes in the
literature.
