It is a big challenge to achieve recording density of 10 Tb/in2 for magnetic recording. Two promising technologies have been proposed for achieving density beyond 1 Tb/in2: heat assisted magnetic recording and bit patterned media. Owing to limitation of minimum media grain size or requirement of higher writing magnetic field, the densities of both technologies are still limited at below 5 Tb/in2. There is a need for new technology to push the density towards 5 Tb/in2 and beyond. Heat assisted bit patterned media recording is a good candidate to achieve this target. In this paper, thermal effects on bit patterned media at densities from 5 to 10 Tb/in2 are studied by numerical simulations. The results show that the media with soft under layer has bit mark vertical expansion of around 0.15 nm at 5 Tb/in2, which may cause recording performance change in the case of less than 3 nm space between magnetic write pole and recording layer. In order to obtain less than 1 ns cooling speed, a pulse heating method is proposed. With this method, the cooling speed gets tremendous improvement at BAR = 1 . Media with large BAR has worse thermal performances.