Energy conditions in $F(T,\mathrm{\Theta })$ gravity and compatibility with a stable de Sitter solution

We study a new type of the modified teleparallel gravity of the form $F(T,\mathrm{\Theta })$ in which $T$, the torsion scalar, is coupled with $\mathrm{\Theta }$, the trace of the stress-energy tensor. In a perturbational approach, we study the stability of the solutions and as a special case we find a condition for stability of the de Sitter phase. Then we adopt a suitable form for $F(T,\mathrm{\Theta })$ that realizes a stable de Sitter solution so that the stability condition creates a specific constraint on the parametric space of the model. Finally, the energy conditions in the framework of $F(T,\mathrm{\Theta })$ gravity is investigated.