We consider an active run-and-tumble particle (RTP) in arbitrary dimension d and compute exactly the probability S(t) that the x-component of the position of the RTP does not change sign up to time t. For the most relevant case of an exponential distribution of times between consecutive tumblings, we show that S(t) is independent of d for any finite time t, as a consequence of the celebrated Sparre Andersen theorem for discrete-time random walks in one dimension. Moreover, we show that this universal result holds for a much wider class of RTP models in which the velocity v of the particle after each tumbling is drawn randomly from an arbitrary probability distribution.