Space plasmas generally posses distribution functions that exhibit high or super thermal energy tails in velocity space which can have different temperatures, a dense cold population and a hot population. Moreover, in laboratory plasmas when a laser or electron beam is passed through a dense plasma, hot low density electron populations can be generated. Presence of such low density electron distributions can act to increase the magnitude of the wave damping rate. In this paper we employ non-Maxwellian distribution function such as the generalized (r, q) distribution function with two electron temperatures to study the Landau damping of electrostatic waves. The results show that the Landau damping increases significantly when the percentage of high energy particles increases and with the increase of the high energy tail and less pronounced shoulders in the profile of the distribution function.