Bicontinuous cubic phases offer advantageous routes to a broad range of applied materials ranging from drug delivery devices to membranes. However, a priori design of molecules that assemble into these phases remains a technological challenge. In this article, high-throughput synthesis of lipidoids which undergo protonation-driven self-assembly (PrSA) into liquid crystalline phases is conducted. With this screening approach we discover 12 different multi-tail lipidoid structures capable of assembling into the bicontinuous double gyroid phase. The large quantity of morphological data uncovers unexpected design criteria that enable phase selection as a function of lipidoid headgroup size and architecture, tail length and architecture and counterion identity. Surprisingly, combining branched headgroups with bulky tails forces lipidoids to adopt unconventional pseudo-disc conformations that pack into double gyroid networks, entirely distinct from other synthetic or biological amphiphiles within bicontinuous cubic phases. From a multitude of possible applications, we demonstrate two examples of functional materials from lipidoid LCs. Firstly, the fabrication of highly aligned lipidoid gyroid films by interfacial PrSA, which are rapidly responsive to the external medium. Secondly, we show that colloidally-dispersed lipidoid cubosomes, e.g. for drug-delivery, are easily assembled using top-down solvent evaporation methods.