  In this work, we have investigated anisotropic neutron stars in the framework
of Rastall-Rainbow gravity. All our calculations were computed using the IU-FSU
realistic equation of state (EoS), in which was considered two cases: standard
nucleonic composition and the one with the eight lightest baryons. From the
neutron star masses and radii obtained we conclude that anisotropic pressure
has significant consequences on the structure of stellar objects. In
particular, when anisotropy is considered within the general relativity
framework, it significantly modifies the maximum stellar mass. On the other
hand, when Rastall-Rainbow gravity and anisotropy are simultaneously
considered, they provide the best results for mass and radius values, including
important astrophysical objects such as the LMXB NGC 6397 and the extremely
massive pulsar millisecond MSP J0740 + 6620. Although the expected inclusion of
hyperons in the nuclear model reproduces stellar masses smaller than those
produced by standard nucleonic matter, we shown that the hyperon puzzle problem
can be solved by including anisotropic effects on compact stars in the context
of the Rastall-Rainbow gravity.
