It is proved by Brooks that any closed orientable 3-manifold with a Heegaard splitting of genus g admits a 2-fold branched cover that is a hyperbolic 3-manifold and a genus g surface bundle over the circle. This paper concerns entropy of pseudo-Anosov monodromies for hyperbolic fibered 3-manifolds. We prove that there exist infinitely many closed orientable 3-manifolds M such that the minimal entropy over all hyperbolic, genus g surface bundles over the circle as 2-fold branched covers of the 3-manifold M is comparable to 1/g.