We discuss grand canonical simulations based on density-functional theory to study the thermodynamic properties of electrochemical interfaces of metallic electrodes in aqueous environments. Water is represented using implicit solvation, here via the self-consistent continuum solvation (SCCS) model, providing a charge-density dependent dielectric boundary. The electrochemical double layer is accounted for in terms of a phenomenological continuum description. It is shown that the experimental potentials of zero charge and interfacial capacitances can be reproduced for an optimized SCCS parameter set [ρmin = 0.0013, ρmax = 0.010 25]. By performing a detailed derivation and analysis of the interface energetics for selected electrochemical systems, we are able to relate the widely used approach of the computational hydrogen electrode (CHE) to a general grand canonical description of electrified interfaces. In particular, charge-neutral CHE results are shown to be an upper-boundary estimate for the grand canonical interfacial free energies. In order to demonstrate the differences between the CHE and full grand canonical calculations, we study the pristine (100), (110), and (111) surfaces for Pt, Au, Cu, and Ag, and H or Cl electrosorbed on Pt. The calculations support the known surface reconstructions in the aqueous solution for Pt and Au. Furthermore, the predicted potential-pH dependence of proton coverage, surface charge, and interfacial pseudocapacitance for Pt is found to be in close agreement with experimental or other theoretical data as well as the predicted equilibrium shapes for Pt nanoparticles. Finally, Cl is found to interact more strongly than H with the interfacial fields, leading to significantly altered interface energetics and structure upon explicit application of an electrode potential. This work underscores the strengths and eventual limits of the CHE approach and might guide further understanding of the thermodynamics of electrified interfaces.