## 1 Introduction  
The discovery of topological insulators [1; 2] stems from the studies on the quantum Hall effect [3; 4] and the quantum spin Hall effects [5; 6; 7; 8; 9] over the past few decades, thus becoming a successful area of research in condensed matter physics. Topological insulators are insulators that have a specific bandgap in bulk form. Nonetheless, unlike ordinary insulators, TIs have metallic Dirac states on the surface [10; 11; 12; 13; 14; 15]. The coexistence of the bulk insulating and surface conducting states leads to TIs with unusual electronic characteristics. For example, TIs are promising materials for spin-dependent applications [2]. So far, several topological materials, such as topological crystalline insulators [16], topological insulators, Dirac [17], and Weyl [18; 19] semimetals have received particular attention and thus have attracted intensive research [1]. It is found that the metallic surface state protected by the time-reversal invariants can always be found in three-dimensional TIs. Furthermore,the metallic surface states of TIs allow only for a specific spin-momentum state, i.e., spin-momentum-locking transport [2]. Therefore, compared with the electron transport of ordinary metals, TIs have great potential for spintronic applications. Double perovskites A\({}_{2}\)BBO\({}_{6}\) made of two single perovskites ABO\({}_{3}\) and AB'O\({}_{3}\) are potential materials for many applications and attract both experimental synthesis and theoretical predictions in the past few decades. For example, Sr\({}_{2}\)FeMoO\({}_{6}\)[20] and Sr\({}_{2}\)FeOsO\({}_{6}\)[21] double perovskites are candidates for spintronic devices for their half metallicity at room temperature. Moreover, Sr\({}_{2}\)CrOsO\({}_{6}\)[22] is a ferrimagnetic insulator with a very high Curie temperature of 720 K. The ideal structure of the double perovskites is cubic, however, they might undergo a temperature-induced structural phase transition. For example, Pb\({}_{2}\)CoTeO\({}_{6}\)[23,24] double perovskites are synthesized with monoclinic, rhombohedral, and cubic structures in different temperature ranges. Beyond the experimental preparations, a lot of theoretical predictions for the emerging double perovskites are made. For example, halfmetallic, topological and multiferroic double perovskites are often mentioned in theoretical predictions. We, therefore, believe that there is possible topological insulator in double perovskites. Experimentally, Bi\({}_{2}\)Se\({}_{3}\)[13,25], Bi\({}_{1-x}\)Sb\({}_{x}\)[26] are shown to be topological insulators. Furthermore, it is theoretically predicted that CsPbI\({}_{3}\)[27,28] and A\({}_{2}\)BiXO\({}_{6}\) (A = Ca, Sr, Ba; X = I, Br) [29] are also possibly TIs. Typically, these materials are ordinary metals or insulators, but become insulators by including spin-orbit coupling (SOC) effect in the calculations. In other words, the topological behavior cannot be discovered without including SOC in theoretical calculations. The CsPbI\({}_{3}\) perovskite is an insulator, it may become a topological insulator by applying an external axial strain. Large bandgap topological insulators are particularly of interest because they have a strong ability to overcome thermal fluctuation. Unlike 2D TIs, where the bandgap in these materials can be tuned in several ways, the bandgap of 3D TIs is hard to increase. Several ways to tune the bandgap of 3D TIs. The first approach is to find a material composed of heavy elements, such as Bi or Te, to provide a stronger SOC [30]. The second approach is to find specific crystal structures, such as perovskite and double perovskite. The last way is to apply external fields. More recently, [PERSON] et al. proposes a new class of large bandgap of 0.55 eV topological insulators in the A\({}_{2}\)BiXO\({}_{6}\) (A = Ca, Sr, Ba; X = I, Br) [30] double perovskites structures. Most recently, [PERSON] et al. also predicts A\({}_{2}\)TePoO\({}_{6}\) (A = Ca, Sr, Ba) [31] double perovskites as TIs which also have high bandgap as high as 0.4 eV.