  In this study, we analyzed the impact of a perfect fluid on the phase
transition of Anti-de Sitter (AdS) black holes within the Rastall gravitational
background. Compared to similar studies in the literature, the findings of this
work are highlighted by the determination of analytical expressions for
critical points for charged and Kerr-Newman AdS black holes using approximated
formulas for the horizon radius. An accurate analysis of these new analytical
expressions allowed us to discover a new viable condition that relates the
Rastall parameter \k{appa}{\lambda} to the equation of state parameter
{\omega}, expressed as: \k{appa}{\lambda} = {\omega}/(1+{\omega}). Thanks to
this new condition, we were able to reproduce all the analytical expressions of
critical points calculated within the framework of Einstein\^as general
relativity for two cases: a charged AdS black hole and a rotating AdS black
hole. These findings suggest that Rastall gravity, considering this new
condition, could serve as an alternative theory of gravitation to general
relativity. The approximate expression of the horizon radius also enabled the
exploration of the distinctiveness of fractional-order phase transitions in
these AdS black holes. Furthermore, we calculated the critical exponents,
offering insights into the behavior of crucial thermodynamic quantities near
the inflection point. Examining how a perfect fluid influences phase transition
reveals various critical behaviors, demonstrating that the variation in the
phase transition depends on the intensity of the perfect fluid. Notably, this
variation is portrayed by a linearly increasing trajectory with the escalation
of this intensity.
