Thermodynamics of accelerating AdS$_4$ black holes from the covariant phase space

We study the charges and first law of thermodynamics for accelerating, non-rotating black holes with dyonic charges in AdS $$_4$$4 using the covariant phase space formalism. In order to apply the formalism to these solutions (which are asymptotically locally AdS and admit a non-smooth conformal boundary $${\mathscr {I}}$$I ) we make two key improvements: (1) We relax the requirement to impose Dirichlet boundary conditions and demand merely a well-posed variational problem. (2) We keep careful track of the codimension-2 corner term induced by the holographic counterterms, a necessary requirement due to the presence of “cosmic strings” piercing $${\mathscr {I}}$$I . Using these improvements we are able to match the holographic Noether charges to the Wald Hamiltonians of the covariant phase space and derive the first law of black hole thermodynamics with the correct “thermodynamic length” terms arising from the strings. We investigate the relationship between the charges imposed by supersymmetry and show that our first law can be consistently applied to various classes of non-supersymmetric solutions for which the cross-sections of the horizon are spindles.