More info would be better, but at first glance a football (round or pointy ends?) sized hot spot at 300°C isn't worth bothering to cool. Continue to run and monitor. Seriously, it isn't all that hot. More of a "warm spot".
Should the spot grow in size or temperature, I'd start with blowing some steam on it to provide cooling. This is very commonly done. Depending on exact circumstances, a sooner than planned shutdown may be in order for a proper refractory or burner repair.