  In this paper, we investigated the thermodynamic properties of Schwarzschild
black hole (SBH) in the non-commutative (NC) gauge theory of gravity. According
to our previous work, we modify the first law of the black hole (BH)
thermodynamics by the physical quantity (NC potential) $\mathcal{A}$ which is
the conjugate to the NC parameter $\Theta$, which leads to this expression
$d\hat{M}=\hat{T}d\hat{S}+\mathcal{A}d\Theta$. Our result shows that the NC SBH
has a phase transition, and the non-commutativity affected this transition. And
the NC potential $\mathcal{A}$ is effective only in the final stage of the BH
evaporation, where it increases the Gibbs free energy at this stage, and the NC
parameter in this study can represent the tension of the spacetime. Then the
study of the pressure of the SBH in the modified first law of the BH
thermodynamic shows a second-order phase transition, and the critical value of
the thermodynamical variables are related to each value of $\Theta$, and that
leads to this parameter to play the same role as a thermodynamical variable.
