The significance of this study lies in its extension of Olsen’s foundational result concerning the classical average box dimension [24], thus broadening its reach and relevance. Specifically, this paper explores the general average box dimensions of typical compact metric spaces (𝒯𝒞ℳs) within the Gromov-Hausdorff space (𝒢𝓗s), utilizing the Gromov-Hausdorff metric (δ𝒢𝓗). We show that the general box dimensions demonstrate significant divergence, even after a “smoothing” effect is applied using sophisticated averaging methods. Finally, our main result is used to reveal that the general box dimensions related to higher-order Hölder and Cesàro averages (h-𝓗𝒞𝒜) exhibit considerable variation.