your run file is doing it wrong, you are using the binary without setting the environment, for that purpose you should use wrappers:
rvm wrapper ruby-2.0.0-p247 --no-links unicorn
To simplify the script use alias so it does not need to be changed when you decide which ruby should be used:
rvm alias create my_app_unicorn ruby-2.0.0-p247
And change the script to:
#!/bin/bash
cd /var/www/html/deploy/production/current
exec 2>&1
exec chpst -u deploy:deploy /home/deploy/.rvm/wrappers/my_app_unicorn/unicorn -E production -c config/unicorn_production.rb
This will ensure proper environment is used for execution of unicorn
and any time you want change ruby used to run it just crate alias to a new ruby.