How much difference it makes on battery usage
Successfully, running nostr-rs-relay (by #[0]) locally on my phone using Termux.

Installation in Termux:
# pkg install rust protobuf
# git clone https://github.com/scsibug/nostr-rs-relay
# cd nostr-rs-relay
# cargo build -r

We can also easily backup and broadcast nostr event using https://nostr-broadcast.pages.dev/ (by #[1])
Discussion
Very little impact on battery. Actually, the most draining usage is my Amethyst not the relay (Termux) 😅