The way in which rain water affects water in a swimming pool depends entirely on the temperature of both elements. As such, there's no universal rule about whether rain water makes swimming pool water warmer or cooler; it all depends on whether the rain is warmer or cooler than the pool.
If the rain is significantly colder than the temperature of your pool water, it will lower your pool water's temperature if it rains for an extended period of time. There's no specific formula that indicates how much the temperature drop will be; it's all dependent on the temperature of the rain, the pool water and how long the rain falls. Cold rain will especially affect the pool's temperature if you use a pool heater to keep the water warm.
If your pool water is cooler than the temperature of the rain, it will warm up during periods of prolonged warm rainfall. Your pool's water may be cool because you've recently filled it with the hose or have had several days of cool weather or even cool rain.
If you're worried about rain affecting the temperature of your pool water, invest in a pool cover to protect the pool water from the rain. While trace amounts of rain may get into the pool, the majority of it will gather on the surface of the pool cover and eventually evaporate without changing the pool's temperature.
To keep your pool a constant temperature, regardless of the temperature and frequency of the rain, use a pool heater. Like the thermostat in your house, you can set the pool heater to any given temperature and it will maintain that temperature. If cool rain cools off the pool water, the heater will bring it back to the level you desire. If warm rain elevates the pool's temperature, the pool heater will stay off until the water's temperature dips below your setting.