  We consider the $SO(4,1)$-covariant fuzzy hyperboloid $H^4_n$ 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 $H^4_n$. We
establish a one-to-one correspondence between the tangential fluctuations of
these spaces.
