Hi! Have the same issue on my 2009 Forester XT. There is a product recall for this issue. https://www.recalls.govt.nz/recall/subaru-forester-impreza-legacy-outback-wrx-model-years-2007-2014 and can melt the pump! I have contacted Armstrong in regards to replacing... waiting to hear back.
Here is the US info https://subaru.oemdtc.com/495/wtm-73-secondary-air-injection-pump-relay-2007-2014-subaru
What does the system do exactly? It helps remove unwanted hydrocarbon emissions, air is injected post combustion for a predetermined amount of time after the engine is started. The system is only operational for a small amount of time post start (the amount of time varies depending on engine temperature, but is typically around 30-60 seconds), and inactive thereafter. It has no effect on engine performance or fuel efficiency it is solely there for emissions control.
Hope this helps!
Cheers
John