  For the quermassintegral inequalities of horospherically convex hypersurfaces
in the $(n+1)$-dimensional hyperbolic space, where $n\geq 2$, we prove a
stability estimate relating the Hausdorff distance to a geodesic sphere by the
deficit in the quermassintegral inequality. The exponent of the deficit is
explicitly given and does not depend on the dimension. The estimate is valid in
the class of domains with upper and lower bound on the inradius and an upper
bound on a curvature quotient. This is achieved by some new initial value
independent curvature estimates for locally constrained flows of inverse type.
