Monolithic Nd: YAG lasers pumped by laser diodes have been widely used in precision measurement fields due to its low frequency noise and intensity noise. Previously, people found that the intensity noise of the pump laser would affect the frequency noise of the Nd: YAG lasers, but the cause has not been well explained so far. In this paper, we propose that this effect is thermally induced. The experimental results agreed well with the theoretical model predictions in up to 60 kHz frequency range. The finding can provide ideas for further reducing the frequency noise of Nd: YAG lasers. We get about 300 Hz linewidth in NPRO laser pumped by single laser diode.