What hardware standards should the flight control computer pass?
In the US for commercial aircraft, SAE-ITC provides recommended design standards in ARINC Characteristic 701-1 Flight Control Computer System (FCCS).  

The recommended environmental standards are in Attachment 6 of ARINC 701. This references environmental categories defined in RTCA DO-160. The issue is that ARINC 701 references an earlier version of DO-160 than the current DO-160G.  

The main reason ARINC 701 has not been kept current (701-1 was the last update in 1983) is that the aircraft manufacturers consider the FCC an integral part of the aircraft and they provide the detail design specifications to their FCC supplier. The specifications typically reference DO-160 (whatever version is current when designing the aircraft). The actual categories and test levels are specified based on their analysis of the environment where the equipment is installed in the aircraft.  

To summarize; the FCC hardware will have to meet the customer's specification which is usually in terms of RTCA DO-160 tests and categories. The actual levels are unique to each aircraft.