Understanding the composition dependence of glass hardness is of critical importance for both advanced glass applications and for revealing underlying fracture mechanisms. We present a topological approach for quantitative prediction of hardness in multicomponent glassy systems. We show that hardness is governed by the number of network constraints at room temperature and that a critical number of constraints is required for a material to display mechanical resistance. Applied to a series of soda lime borate glasses, the predicted values of hardness are in excellent agreement with experimental measurements. Our approach is generally applicable to any network glass and demonstrates the importance of accounting for the temperature dependence of the network constraints.