Due to their future applications in optical communication, nonlocal dark solitons in bulk medium and surface bright solitons have received much attention recently. However, nonlocal surface dark solitons have not been investigated till now. In this paper, 1+1 dimensional nonlocal fundamental and second-order surface dark solitons have been found numerically at the interface between thermal nonlinear medium and linear medium. The relation between the wave shape of nonlocal surface dark soliton and propagation constant and nonlocality degree is investigated. Moreover, the stability of them is analyzed theoretically. The numerical simulation results show that 1+1 dimensional nonlocal fundamental surface dark Solitons are always stable in the domain of their existence, while second-order surface dark solitons are oscillatorily unstable and the width of unstable domain depends more greatly on propagation constant than nonlocality degree of nonlocal nonlinear medium. The figure showing the propagation, with the initial input of noise added, confirms the correctness of stability analysis results.