▎ 摘　　要

In this paper, we consider the particle on curved graphene space-time. In that case, we calculate the geometric form of potential which is known as Gaussian function. Here, we introduce the metric background which completely corresponds to curved graphene space-times. This metric leads us to obtain the geometry potential and we make the Laplace Beltrami equation in the mentioned metric background. We also rearrange such relation in terms of the second-order equation. By using the known polynomial, we solve the particle equation of motion in graphene background. In that case, we arrive the energy spectrum which has three terms. We take advantage from energy spectrum and investigate the thermal properties of system. The additional terms give us an opportunity to obtain the corrected entropy and free energy. So, we show that the additional term comes from geometry potential. This correction is important for the large scale. Hence, we show that correction term is logarithmic as well as small scale corrections.