Temporal logic plays a crucial role in specifying and reasoning about dynamic systems, where temporal constraints and properties to be monitored are essential. Traditional approaches like LTL-monitoring assume monotonicity, which limits their applicability to scenarios involving non-monotonic temporal properties. We delve into complexity aspects of monitoring temporal specifications using non-monotonic Temporal Equilibrium Logic (TEL), a temporal extension of Answer Set Programming defined over Temporal Here and There Logic (THT) with a minimality criterion enforcing stable models. Notably, we study the complexity gap between monitoring properties in THT and TEL semantics, and the complexity of monitoring approximations based on progression, which is widely used in verification and in AI. In that, we pay particular attention to the fragment of temporal logic programs.