From social interactions to the human brain, higher-order networks are key to describe the underlying network geometry and topology of many complex systems. While it is well known that network structure strongly affects its function, the role that network topology and geometry has on the emerging dynamical properties of higher-order networks is yet to be clarified. In this perspective, the spectral dimension plays a key role since it determines the effective dimension for diffusion processes on a network. Despite its relevance, a theoretical understanding of which mechanisms lead to a finite spectral dimension, and how this can be controlled, represents nowadays still a challenge and is the object of intense research. Here we introduce two non-equilibrium models of hyperbolic higher-order networks and we characterize their network topology and geometry by investigating the interwined appearance of small-world behavior, $delta$-hyperbolicity and community structure. We show that different topological moves determining the non-equilibrium growth of the higher-order hyperbolic network models induce tunable values of the spectral dimension, showing a rich phenomenology which is not displayed in random graph ensembles. In particular, we observe that, if the topological moves used to construct the higher-order network increase the area$/$volume ratio, the spectral dimension continuously decreases, while the opposite effect is observed if the topological moves decrease the area$/$volume ratio. Our work reveals a new link between the geometry of a network and its diffusion properties, contributing to a better understanding of the complex interplay between network structure and dynamics.
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