There is this new awesome map Proving Ground.
In this game you start with enough gold to buy the Philosopher's stone. But the tooltip for earned money starts counting without actually getting the money.
Is this a bug or do I get the money later?
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Contrary to what others have stated, gold per 10 does in fact work on proving grounds. In Classic (Summoner's Rift/Twisted Treeline) the champion passively earns 13 gold per 10 seconds, in Dominion/Proving Grounds the passive gold is 56 gold per 10 seconds. This number is increased not only by gold per 10 items but also runes/masteries that have the same effect.
Also ARAM has nothing to do with the effectiveness of the stat.
Now about your actual question: You won't get the money later. The tooltip may inaccurately represent the amount you're getting before the gold starts actually rolling in when minions spawn. The total gold earned part of the tooltip is relatively new and Riot is known to have some bugs scattered around all over the place. But as far as I know the tooltip is accurate once minions spawn.
GP10 Items do not work on this map since you already have increaased passive gold gain. Philo Stone and other gp10 items are still available tho cause they 1. give some decent stats 2. are upgraded into other items.
In AllRandomAllMid mode (which is the name used for the Proving Grounds Map) all the Items that provide static income in the normal game no longer do so.
Riot created this map with ARAM in mind. As such, they saw fit to give everyone passive Gold per 10 and mana regen, with heals being half as effective to make the games go by more quickly.
In addition to that, they removed Wards and some AD items (The Bloodthirster and Madred's Bloodrazor being a couple), as well as making GP/10 item unique passives count against the map passive. Their mana/health regen/ health/critchance/AP is still given, but the UP GP/10 no longer do on that map.