Towards the emergence of non-zero thermodynamical quantities for Lanczos-Lovelock black holes dressed with a scalar field

The present work aims to explore the model given by Lanczos-Lovelock gravity theories indexed by a fixed integer to require a unique anti-de Sitter vacuum, dressed by a scalar field non-minimal coupling. For this model, we add a special matter source characterized by a non-linear Maxwell field coupling with a function depending on the scalar field. Computing its thermodynamics parameters by using the Euclidean action, we obtain interesting and non zero thermodynamical quantities, unlike its original version, allowing analyzing thermodynamical stability. Together with the above, we found that these solutions satisfy the First Law of Thermodynamics as well as a Smarr relation.