Energy loss of a heavy particle near 3D charged rotating hairy black hole

In this paper, we start with a black brane and construct a specific space-time which violates hyperscaling. To obtain the string solution, we apply the Null-Melvin Twist and KK reduction. Using the difference action method, we study the thermodynamics of the system to obtain a Hawking–Page phase transition. To have hyperscaling violation, we need to consider $\theta =\frac{d}{2}.$ In this case, the free energy $F$ is always negative and our solution is thermal radiation without a black hole. Therefore, we find that there is no Hawking–Page transition. Also, we discuss the stability of the system and all thermodynamical quantities.