If you're still encouraging the new grass through to establishment then the use of germination sheets will keep the frost off and the ground insulated. The only problem associated with the use of these covers is the risk of disease. Remove the covers during the day to minimise the risk, but spray with a fungicide if disease is spotted.
As the weather closes in there is a risk that moss will now develop, the wetter weather makes it almost inevitable. Again the use of an approved moss killer will check any growth quickly. Once the moss has turned and it becomes dry enough, scarify the affected areas to remove the dying vegetation. One of the difficulties with spraying chemicals at this time of year is getting an accurate forecast to know that there is a dry window of opportunity. The last thing you need is to spray a chemical for it to be rendered ineffective by rain, apart from the fact that these products are expensive to buy in the first place.
It may be possible to mow the square once a week, depending on growth and soil conditions; retain the cutting height at 25mm. Also in dry conditions a monthly slit or solid tine aerating operation will help to get air to the roots and allow quicker water dispersion.
Try to remove dew from the square, daily if at all possible; with the warm soils and damp conditions, disease will be quick to spread through a blanket of moisture.
The frosty nights will have brought autumn leaves down in abundance, make sure they are raked up and collected, as piles of leaves will also help harbour disease on the turf.