The extensive densification of mobile networks is increasing the network energy consumption and leading to remarkable economical and sustainability concerns. At the same time, regulatory and physical constraints, especially in urban environments, may limit the network expansion and the free installation of Base Stations (BSs). In this context, High Altitude Platform Stations (HAPSs) are emerging as a promising solution to host aerial BSs that can provide additional capacity over a wide geographical area, to offload the on-ground mobile network and support a sustainable transition towards the 6G era. This paper investigates the potential of HAPS offloading to reduce the energy demand from the grid and the operational cost of mobile networks. Our results highlight the effectiveness of HAPS offloading in reducing the size of the RE supply that is required to achieve grid energy reduction on the terrestrial network, thus enhancing the feasibility of a sustainable evolution towards 6G networks. Different allocation strategies are designed and analyzed under several configuration settings, to dynamically adapt the HAPS capacity to the traffic variability in space and over time. A fine tuning of the strategy settings is proved effective in trading off physical constraints, operational cost, sustainability goals, and Quality of Service.