We show that Einstein’s field equations with a negative cosmological constant
can admit black hole solutions whose thermodynamics coincides with that of
logotropic fluids, recently investigated to heal some cosmological and
astrophysical issues. For this purpose, we adopt the Anton-Schmidt equation of
state, which represents a generalized version of logotropic fluids. We thus
propose a general treatment to obtain an asymptotic anti-de Sitter metric,
reproducing the thermodynamic properties of both Anton-Schmidt and logotropic
fluids. Hence, we explore how to construct suitable spacetime functions,
invoking an event horizon and fulfilling the null, weak, strong and dominant
energy conditions. We further relax the strong energy condition to search for
possible additional solutions. Finally, we discuss the optical properties
related to a specific class of metrics and show how to construct an effective
refractive index depending on the spacetime functions and the thermodynamic
quantities of the fluid under study. We also explore possible departures with
respect to the case without the fluid.
