6G networks aim to facilitate the global sustainability drive. However, sustainability of 6G networks itself will pose major research questions due to the increasing number of devices, systems, and services that can exacerbate the consumption of key resources. In this work, we focus our investigation to 6G security, since sustainability is mostly overlooked when it comes to security. Emerging security technologies which are highly dependent on developing concepts in the realm of security such as AI and blockchain require huge amounts of resources. This article aims to stir further research on sustainability of 6G security through highlighting the main research challenges and future research directions in this domain.