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The solutions for the Tolmann-Oppenheimer-Volkoff (TOV) equation bringvaluable informations about the macroscopical features of compact astrophysicalobjects as neutron stars. They are sensitive to both the equation of stateconsidered for nuclear matter and the background gravitational theory. In thiswork we construct the TOV equation for a conservative version of the $f(R,T)$gravity. While the non-vanishing of the covariant derivative of the $f(R,T)$energy-momentum tensor yields, in a cosmological perspective, the prediction ofcreation of matter throughout the universe evolution as shown by T. Harko, inthe analysis of the hydrostatic equilibrium of compact astrophysical objects,this property still lacks a convincing physical explanation. The imposition of$\nabla^{\mu}T_{\mu\nu}=0$ demands a particular form for the function $h(T)$ in$f(R,T)=R+h(T)$, which is here derived. Therefore, the choice of a specificequation of state for the star matter demands a unique form of $h(T)$,manifesting a strong connection between conserved $f(R,T)$ gravity and the starmatter constitution. We construct and solve the TOV equation for the generalequation of state for $p=k\rho^{\Gamma}$, with $k$ being the EoS parameter,$\rho$ {\it the energy density} and $\Gamma$ is the adiabatic index. We alsoderive the macroscopical properties of neutron stars ($\Gamma=5/3$) within thisapproach.

A conservative energy-momentum tensor in the $f(R,T)$ gravity and its implications for the phenomenology of neutron stars