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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. The recently proposed Black Hole Entropy Distance Conjecture are extended 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, the authors 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, the evaporation of these black holes is studied and related to the geometric distance. It is found that the corresponding distance to the final stage of evaporation is finite. This leads to the conclusion that evaporation does not lead to the appearance of a light tower of black hole microstates. [ABSTRACT FROM AUTHOR]