We set the goal to study the properties of invariant submanifolds of the hyperbolic Sasakian manifolds. It is proven that a three-dimensional submanifold of a hyperbolic Sasakian manifold is totally geodesic if and only if it is invariant. Also, we discuss the properties of η-Ricci-Bourguignon solitons on invariant submanifolds of the hyperbolic Sasakian manifolds. Finally, we construct a non-trivial example of a three-dimensional invariant submanifold of five-dimensional hyperbolic Sasakian manifold and validate some of our results