We propose a variant of the Rapidly Exploring Random Tree Star (RRT*) algorithm for planning trajectories of linear systems under input constraints subject to spatio-temporal specifications expressed in a fragment of Signal Temporal Logic (STL). Compared to existing sampling-based approaches that rely on mixed-integer or non-smooth optimization, which suffer from poor scalability, we adopt a control-theoretic framework based on set forward invariance. First, STL specifications with polyhedral predicates are encoded as a time-varying set computed via linear programming, such that all trajectories evolving within the set satisfy the specification. Forward invariance of the set for linear systems under input constraints is guaranteed using non-smooth analysis. Second, a modified RRT* algorithm is proposed to efficiently sample dynamically feasible and minimum cost trajectories within this time-varying set, in order to satisfy the given specifications. The effectiveness of the approach is demonstrated on an autonomous inspection mission of the International Space Station and a timed room-servicing task.