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We present a generalization of Rastall's gravity in which the conservationlaw of the energy-momentum tensor is altered, and as a result, the trace of theenergy-momentum tensor is taken into account together with the Ricci scalar inthe expression for the covariant derivative. Afterwards, we obtain the fieldequations in this theory and solve them by considering a spherically symmetricspace-time. We show that the external solution has two possible classes ofsolutions with spherical symmetry in the vacuum in generalized Rastall'sgravity, and we analyse one of them explicitly. The generalization, in contrastto constant value $k=8\pi G$ in general relativity, has a gravitationalparameter $k$ that depends on the Rastall constant $\alpha$. As an application,we perform a careful analysis of the effects of the theory on neutron starsusing realistic equations of state (EoS) as input. Our results show thatimportant differences on the profile of neutron stars are obtained within tworepresentatives EoS.

Generalized Rastall's gravity and its effects on compact objects