Identification of the geometric structures of silver clusters is of great importance in future nanotechnologies due to their superior properties. Nevertheless, some ground-state structures are still in academic debate, partly because the experiments and theoretical calculations are not performed at the same temperatures. For example, silver clusters usually have compact configurations. However, a combined experimental and theoretical study proposed that the most stable structure of Ag-13- had a two-coordinated atom. By using the CALYPSO approach for the global minima search followed by first-principles calculations, we discovered that a more compact trilayer Ag-13- cluster was the ground state, in accordance with another three works published recently. In addition, its O-2 adsorption structure is also energetically favored. By tracing characteristic bond changes in ab initio molecular dynamics (MD) simulations, we confirmed that, compared with other isomers, this trilayer structure and its O-2 adsorption structure also had the highest thermal stability. This work emphasized the thermal stability concept in theoretical calculations, which may be a necessary supplement to explain the experimental observations on cluster science.