If you rely on the room temperature to be warm enough, then cool weather can definitely have an effect. Temperatures in the 50's through the lower 70's can cause turtles to become lethargic and stop eating. They are not hibernating, and this is a risky state for them to be in. They often become ill when even a little bit chilly. The water should have submersible aquarium heater in it which will maintain a temperature around 78*F. The basking area should be 85*F to 90*F.
To help you check to make sure everything is right, I'm sending along a slider care sheet I've written. You can use it as a checklist.SLIDER TURTLE CARE SHEET
Well-cared for sliders can live 30 years or more.The Tank
It's recommended that a baby slider have at least a 15 gallon tank. By the time the turtle is 3-4 years old, it will need a 60 gallon tank, so it's best to get the biggest you can in the beginning. You can also use a large RubberMaid tote. That's not as pretty as a tank, but costs a lot less. Set up the tank so there's a land area and a water area. Put the basking light at one end so the whole tank doesn’t get too hot. You want the water to be about twice as deep as the turtle is long. If the turtle is two inches long, you'll want four inches of water.Temperatures and Basking Area
Turtles need certain types of lighting and need to be warm. Air and water that are not warm enough can lead to fungal and respiratory infections and unhealthy shells. Turtles must have a basking area where they can get out of the water, dry off, and bask in very warm light. The ambient air temperature in the tank should be around 75 *F (24*C) , with the basking area warmer still. Over the basking area there should be some sort of lamp that will take a 40-60 watt incandescent bulb (or you can buy a ceramic light fixture made just for reptiles). If you live in an area that has farm stores, you can buy a metal light fixture made to keep baby chicks warm for just a few dollars. Don't buy the accompanying bulb, however. You need an ordinary incandescent bulb in the basking light. Hardware stores sell similar fixtures as work lights. The basking area should be kept at 85-90*F (29 to 32*C). Use a digital probe thermometer to be sure. You can adjust the temperature by raising or lowering the light fixture.The lights that come with the covers on aquariums are not suitable for turtles. You'll also need a submersible aquarium heater that will keep the water 78-82*F (26 to 28*C).UVB Light
It's extremely important that you buy an additional light that produces UVB rays. A Reptisun 10.0 is a good brand that does. If you choose another brand be absolutely certain it provides UVB rays. Don't take the word of pet store personnel, but read it for yourself. Full-spectrum, DayGlo, SunGlo, UV, or UVA are not the same thing. I'm putting a lot of emphasis on this because it's crucial to your turtle's health. Without this light, Metabolic Bone Disease (MBD) will develop because your turtle won't be able to produce vitamin D. Vitamin supplements are not a good replacement for the proper lighting. MBD causes a very slow and painful death.UVB bulbs must be replaced every six months as they lose their effectiveness after that, even though they may still look fine. Light that comes through a window isn't sufficient because the glass filters out most of the rays turtles need to stay healthy. To prevent MBD, turtles also need calcium. The easiest way to provide it is to place a cuttlebone in the tank. Cuttlebones are sold in bird departments of pet stores.Filtration
Turtles are very sensitive to water quality. Even if you change the water every day, it can still contain harmful chemicals. A good filtration system is essential. Water changes are also needed even with a filter. If the tank is too small, no filter can keep up with the amount of waste that turtles produce.Feeding
Feeding is an area where pet stores often give out bad information. Commercial food should make up only 1/4 of the diet. Animal products (cooked meat, earthworms, canned cat food) should make up another 1/4. The remaining half should be plant foods (dark lettuce like romaine, bits of strawberry or melon, etc.). Hatchlings should be fed every day. Older turtles should be fed 3 times per week. Overfeeding can lead to gout and kidney failure.
For Further Reading
This is among the most reputable sites on turtles.