  We provide sharp stability estimates for the Alexandrov Soap Bubble Theorem
in the hyperbolic space. The closeness to a single sphere is quantified in
terms of the dimension, the measure of the hypersurface and the radius of the
touching ball condition. As consequence we obtain a new pinching result for
hypersurfaces in the hyperbolic space.
  Our approach is based on the method of moving planes. In this context we
carefully review the method and we provide the first quantitative study in the
hyperbolic space.
