RTCA DO-160, Environmental Conditions and Test Procedures for Airborne Equipment, is the standard test specification for the environmental qualification of all airborne equipment, including battery systems. The current edition is DO-160G.
Battery systems installed on aircraft must demonstrate compliance with the relevant sections of DO-160 as part of their airworthiness approval. Key sections applicable to batteries include: temperature and altitude, temperature variation, humidity, vibration, mechanical shock, magnetic effect, power input, voltage spike, audio frequency conducted susceptibility, induced signal susceptibility, radio frequency susceptibility, and fire, flammability and fluid susceptibility.
References: RTCA DO-160 (official document page)
See also: DO-311A