Ice on the inside. Same as a hot drink warms you in winter. Same principle for ice lollies and icy drinks in the summer.
I lived in a non-air-conditioned house in Sydney a decade or so ago, and the fans weren't much use when it got above 40. I learned to always, always keep frozen berries and juice in the fridge to make smoothies, and to buy bagfuls of ice, and to keep tap water in bottles in the fridge. If you fill and leave on the side for a few minutes with the top off, the chlorine evaporates, and then when chilled it tastes like any other bottled water.
Right now I have two bags of ice, lots of home-bottled water, and about 20 ice lollies in the freezer (Pip ones, and Jude's milk lollies, which are very low in sugar and calories and, as small, come in boxes of six so last a lot longer), as well as frozen smoothie mixes and chilled juice. Lots of chilled apples and satsumas in the fridge as snacks for the kids, too.
I also draw the curtains through the midday on any windows that get full sun, and open all windows I can as soon as it starts to get dark.
Oh, and I can't sleep without covers, either. So I just use a sheet in hot weather.