I have recently had my phone laying on my desk with the game active. I noticed that there spawn quite a lot of pokemons around me. Is this because I have the game active, or would I get the same result without having the game active (and often opening the game to catch them)?
As RenaissanceProgrammer said in the comments, it's more likely that you're just noticing it more when you have it open.
I've been doing the same thing - leaving it open on my desk at work, and occasionally a new Pokemon will spawn.
Likewise, at home I've been opening it every now and then. Sometimes there's a Pokemon already nearby, other times there's not.
But, comparing these two scenarios, I haven't gotten any large number of additional Pokemon when I've left it open. (Having said that, I've found leaving it open brings other benefits, like adding more GPS-fix 'steps' to help you hatch eggs).