  We propose a uniform block diagonal preconditioner for the condensed
$H$(div)-conforming HDG scheme of a parameter-dependent saddle point problem
that includes the generalized Stokes problem and linear elasticity. An optimal
preconditioner is obtained for the stiffness matrix for the
velocity/displacement block via auxiliary space preconditioning (ASP)
technique. A robust preconditioner spectrally equivalent to the Schur
complement of element-piecewise constant pressure space is also constructed.
Finally, numerical results of generalized Stokes and steady linear elasticity
equations verify the robustness of our proposed preconditioner with respect to
mesh size, Lam\'e parameters and time step size.
