The first was played at Meggetland against Leinster while last time out - before the Rugby World Cup break - they travelled to Treviso and won 27-24.
But Grant is looking forward to the Ospreys clash at BT Murrayfield next Friday and believes home form could prove crucial for the Scottish side's hopes.
"Hopefully we can make it habitual and get on a good roll and get a good turnout for Ospreys," said Grant.
"That'll make such a huge difference, if supporters can get behind us and get along because it's a massive old game for us now.
"We'll enjoy the break, it's been a massive four games - we'll hopefully fire into them [Ospreys] and muster up as much support as we can."
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Guinness PRO12 Edinburgh News
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