[en] We consider the $SO(4,1)$-covariant fuzzy hyperboloid $H_n^4$ as a solution of Yang–Mills matrix models, and study the resulting higher-spin gauge theory. The degrees of freedom can be identified with functions on classical $H^4$ taking values in a higher-spin algebra associated to $\mathfrak{so}(4,1)$, truncated at spin n. We develop a suitable calculus to classify the higher-spin modes, and show that the tangential modes are stable. The metric fluctuations encode one of the spin 2 modes, however they do not propagate in the classical matrix model. Gravity is argued to arise upon taking into account induced gravity terms. This formalism can be applied to the cosmological FLRW space-time solutions of [1], which arise as projections of $H_n^4$. We establish a one-to-one correspondence between the tangential fluctuations of these spaces.