Using the internal temperature sensor of the AVR chip.
Most new AVR chips (used in the Arduino) have an internal temperature sensor. It is not often used, since it is not accurate. There are however a number of situations that this sensor can be used.
AVR chips that have an internal temperature sensor:
The internal temperature is the temperature inside the chip, just like the cpu-temperature of a computer. If the Arduino is not sleeping, this temperature will increase. If output pins are used to drive current (for example to drive a led) the internal temperature increases more.
This internal temperature can not be used to read the ambient temperature.
The ambient temperature could be measured if the Arduino turns on, but only if the Arduino was off for more than 10 minutes.
The internal temperature can also be used if the normal operating temperature and the ambient temperature is known. Suppose the internal temperature would raise 5 degrees during normal operation. An increase of 20 degrees would indicate a problem.
In situations with high temperatures a calibrated temperature reading could prevent damage. Most newer AVR chips have a temperature range up to 85 degrees Celsius. The Arduino could be used to switch off the device or to switch off itself at 80 degrees Celsius.
According to the datasheet, the temperature could be off by 10 degrees Celsius. But an accuracy of about 2 degrees Celsius is possible if the gain and offset is measured.
For accurate temperatures, every chip should be calibrated. If a few Arduinos of the same type are used, the internal temperature sensor still could differ.
The gain and offset is different for the different types. It is also different for older and newer chips.
Use this to start with for the ATmega328 types (temperature in degrees Celsius):
temperature = (ADCW - 324.31) / 1.22
If the Arduino has not been used for half an hour, turn it on, and check the internal temperature. If there is a difference with the room temperature, adjust the offset in the program. After that, don't change it anymore.
For more accuracy the gain and offset has to be determined by measuring the sensor values for a few temperatures.
The internal temperature might increase by a few degrees Celsius after about 10 minutes. If there is voltage controller near the AVR chip, the internal temperature will increase more. An increase of 10 degrees Celsius is normal.
To check the circuit board where the heat is generated, an infrared temperature meter (non-contact IR thermometer) can be used to check all components.
The temperature sensor is at MUX channel 8, and must be used with the internal reference. With Arduino 1.0 (2012) channel 8 can not be selected with the library functions.