  We study the thermodynamic properties of the Reissner-Nordstr\"om black hole
with cosmological constant, expressed in terms of the curvature radius, by
using the approach of shadow thermodynamics and the formalism of
geometrothermodynamics. We derive explicit expressions for the shadow radius in
terms of the horizon, photon sphere, and observer radii. The phase transition
structure turns out to strongly depend on the value of the curvature radius,
including configurations with zero, one, or two phase transitions. We also
analyze the black hole microscopic structure and find differences between the
approaches of thermodynamic geometry and geometrothermodynamics, which are due
to the presence of the curvature radius. We impose the important condition that
the black hole is a quasi-homogeneous thermodynamic system to guarantee the
consistency of the geometrothermodynamic approach.
