  We extend the recently proposed Black Hole Entropy Distance Conjecture to the
case of charged black holes in de Sitter space. By systematically studying
distances in the space of black hole geometries with multiple horizons, we find
that the distance is generically related to the logarithm of the entropy. From
the infinite distance conjecture this predicts the appearance of a massless
tower of modes in the limit of infinite entropy. Further, we study the
evaporation of these black holes and relate it to the geometric distance. We
find that the corresponding distance to the final stage of evaporation is
finite. We conclude that evaporation does not lead to the appearance of a light
tower of black hole microstates.
