Thermodynamics of a Static Dilaton Black Hole

The thermodynamics of a static dilaton black hole is investigated by Euclidean path integral. Four interesting results are obtained: (I) Although the temperature, heat capacity and chemical potential have different forms as compared to general classical static charged black hole, the Bardeen-Carter-Hawking thermodynamical laws are valid for the black hole. (II) The entropy and area of the event horizon are zero but the energy, temperature and surface gravity are non-zero for extremal case ( r_h = 2M = b). (III) The charged black hole does not possess second order phase transitions as the Reissner-Nordström black hole since its heat capacity is negative. (IV) The action is equal to the entropy, and both quantities coincide with one quarter of area of the event horizon.