Electrosorption of solvated species at metal electrodes is a most fundamental class of processes in interfacial electrochemistry. Here, we use its sensitive dependence on the electric double layer to assess the performance of ab initio thermodynamics approaches increasingly used for the first-principles description of electrocatalysis. We show analytically that computational hydrogen electrode calculations at zero net-charge can be understood as a first-order approximation to a fully grand canonical approach. Notably, higher-order terms in the applied potential caused by the charging of the double layer include contributions from adsorbate-induced changes in the work function and in the interfacial capacitance. These contributions are essential to yield prominent electrochemical phenomena such as non-Nernstian shifts of electrosorption peaks and non-integer electrosorption valencies. We illustrate this by calculating peak shifts for H on Pt electrodes and electrosorption valencies of halide ions on Ag electrodes, obtaining qualitative agreement with experimental data already when considering only second order terms. The results demonstrate the agreement between classical electrochemistry concepts and a first-principles fully grand canonical description of electrified interfaces and shed new light on the widespread computational hydrogen electrode approach.