  We explore gravity-independent equations of state for neutron stars,
particularly focusing on twin stars. Examining four categories, we emphasize
their behavior in both General Relativity and Palatini gravity. Additionally,
we discuss a subcategory of type I, which, in the context of General
Relativity, does not exhibit twin star phenomena, yet demonstrates this
phenomenon in modified gravity. Furthermore, we briefly address challenges
associated with the negative trace of the energy-momentum tensor, prevalent in
both theories.
