  The present article is devoted to constrain the model parameter $\chi$ for
the $f(R,T)= R + \chi T$ gravity model by employing the constraints coming from
big bang nucleosynthesis. We solve the field equations and constrain $\chi$ in
the range $-0.14 \kappa^{2} \leq \chi \leq 0.84 \kappa^{2}$ (where $\kappa^{2}
= \frac{8 \pi G}{c^{4}}$) from the primordial abundances of light elements such
as helium-4, deuterium and lithium-7. We found the abundances of helium-4 and
deuterium agrees with theoretical predictions, however the lithium problem
persists for the $f(R,T)$ gravity model. We also investigate the evolution of
entropy for the constrained parameter space of $\chi$ for the radiation and
dust universe. We report that entropy is constant when $\chi = 0$ for the
radiation dominated universe, whereas for the dust universe, entropy increases
with time. We finally use the constraints to show that $\chi$ has negligible
influence on the cold dark matter annihilation cross section.
