Thermodynamic analysis and shadow bound of black holes surrounded by a dark matter halo

We perform the thermodynamic analysis of a black hole (BH) immersed in a dark matter halo (DMH). It is shown that the BH could not be in thermal equilibrium with the DMH in any regions outside the event horizon. This means that the thermodynamic influence of the environment (DMH) is relatively small on the BH. Importantly, it does not alter the nature of the negative heat capacity for the BH. We stress that the Newtonian \((1/a_0)\) approximation gives us a correct thermodynamic description for the BH surrounded by DMH because the first law of thermodynamics and Smarr formula are satisfied. Hence, the Newtonian Helmholtz free energy is employed to reveal that there is the absence of phase transition to other BH with a positive heat capacity. Finally, we investigate the shadow bound of favored region for the BH immersed in the DMH by comparing EHT observations.