Summary: Considering the anisotropic Navier-Stokes equations as well as the primitive equations, it is shown that the horizontal velocity of the solution to the anisotropic Navier-Stokes equations in a cylindrical domain of height \(\varepsilon\) with initial data \(u_0=(v_0,w_0)\in B_{q,p}^{2-2/p}\), \(1/q+1/p\leq 1\) if \(q\geq 2\) and \(4/3q+2/3p\leq 1\) if \(q\leq 2\), converges as \(\varepsilon\rightarrow 0\) with convergence rate \(\mathcal{O}(\varepsilon)\) to the horizontal velocity of the solution to the primitive equations with initial data \(v_0\) with respect to the maximal-\(L^p-L^q\)-regularity norm. Since the difference of the corresponding vertical velocities remains bounded with respect to that norm, the convergence result yields a rigorous justification of the hydrostatic approximation in the primitive equations in this setting. It generalizes in particular a result by J. Li and E. S. Titi [J. Math. Pures Appl. (9) 124, 30–58 (2019; Zbl 1412.35224)] for the \(L^2-L^2\)-setting. The approach presented here does not rely on second order energy estimates but on maximal \(L^p-L^q\)-estimates which allow us to conclude that local in-time convergence already implies global in-time convergence, where moreover the convergence rate is independent of \(p\) and \(q\).