Due to its future use in communication area, nonlocal spatial optical soliton has been a hot research topic recently. However, because of its special border condition, little research has been done on spatial dark solitons especially on its linear stability. In this paper, a method to analyze linear stability of nonlocal spatial dark soliton is put forward, moreover a numerical simulation and analysis is done on the linear stability of (1+1)-dimensional fundamental and second-order dark soliton in thermal nonlinear medium. Numerical results show that (1+1)-dimensional fundamental nonlocal dark solitons are always stable in their entire existence domain, while second-order dark solitons are oscillatorily unstable and the width of unstable domain depends on propagation constant and nonlocality degree of thermal nonlinear medium. The propagation graphs of initial input with noise addition confirms the correctness of linear stability analysis results.