In this paper, we study fluctuations of the volume of a stable sausage defined via a d-dimensional rotationally invariant α-stable process. As the main results, we establish a functional central limit theorem (in the case when d/α > 3/2) with a standard one-dimensional Brownian motion in the limit, and Khintchine's and Chung's laws of the iterated logarithm (in the case when d/α > 9/5).