<p>We show that the existence of constant scalar curvature Kähler (cscK) metrics with cone singularities is equivalent to the properness of log <i>K</i>-energy. We also prove their equivalence to the log geodesic stability. Our results extend the work of Chen–Cheng to the setting of cscK cone metrics, where they prove the properness conjecture and Donaldson’s geodesic stability conjecture for smooth Kähler metrics. As an application, we prove an approximation property for the path of cscK cone metrics. That generalises Donaldson’s continuity method, which utilised a path of Kähler–Einstein (KE) cone metrics to resolve the Yau-Tian-Donaldson conjecture for Fano KE metrics.</p>