  In this work we explore the connections between Newman-Penrose scalars,
including the Penrose-Rindler $\mathcal{K}$-curvature, with the equation of
state of asymptotically Anti-de Sitter Reissner-Nordstr\"om black holes. After
briefly reviewing the equation of state for these black holes from the point of
view of both the Extended Phase Space and the Horizon Thermodynamics
approaches, a geometric splitting is given for such an equation in terms of the
non vanishing Newman-Penrose scalars which define the $\mathcal{K}$-curvature
at the horizon. From this splitting, a possible thermodynamical interpretation
is developed for such scalars in the context of the black hole thermodynamics
approaches initially discussed. Afterwards, the square root of the Bel-Robinson
tensor is employed to propose conditions at the horizons in terms of pressures
or energy densities, which can be understood as alternative thermodynamical
definitions of these surfaces.
