A Sustainability Roadmap for C$^3$

The particle physics community has agreed that an electron-positron collider is the next step for continued progress in this field, giving a unique opportunity for a detailed study of the Higgs boson. Several proposals are current under evaluation of the international community. Any large particle accelerator will be an energy consumer and so, today, we must be concerned about its impact on the environment. This paper evaluates the carbon impact of the construction and operations of one of these Higgs factory proposals, the Cool Copper Collider (C$^{3}$). It introduces several strategies to lower the carbon impact of the accelerator. It proposes a metric to compare the carbon costs of Higgs factories, balancing physics reach, energy needs, and carbon footprint for both construction and operations, and compares the various Higgs factory proposals within this framework. For C$^{3}$, the compact 8 km footprint and the possibility for cut-and-cover construction greatly reduce the dominant contribution from embodied carbon.