No one seems to have answered your question so i will give you at least a bit of what you are asking for.
I don't believe there is really any way of suspending it automatically depending on what work space it is on. Or that it always be a good idea. What if you are streaming music or downloading a file? The TCP connection will hang and the server drop it after a while.
There are commands you can use that might get you in the right direction of "semi-automation"
running the command:
kill -STOP pidOfProgram
will background / stop the process.
kill -CONT pidOfProgram
will foreground / return it to its previous state
Pids change though. If you want to use it in a script (so you can bind it to a key combination for example)
you could try using use this:
kill -STOP `pgrep -o chrome`
pidof -o chrome