We study some thermodynamical quantities solving the Klein-Gordon equation by using a potential combining Coulomb, and Woods-Saxon potentials for a particular spatially dependent mass form. We find the energy eigenvalues, and the corresponding wave functions approximately by reforming the Klein-Gordon equation similar to a Riemann-type equation whose solutions are given in terms of hypergeometric function (2)F1(p', q'; r'; z). After obtaining them analytically, we get the partition function which is based on studying the thermostatistical quantities such as specific heat and Tsallis entropy. As a function of temperature, the specific heat shows a maximum structure which is the well-known Schottky anomaly at very low temperature. Tsallis entropy enhances with increasing the temperature for different surface thickness and width of the potential.