We measure the low-frequency thermal fluctuations of pure spin current in a platinum film deposited on yttrium iron garnet via the inverse spin Hall effect (ISHE)-mediated voltage noise as a function of the angle α between the magnetization and the transport direction. The results are consistent with the fluctuation-dissipation theorem in terms of the recently discovered spin Hall magnetoresistance (SMR). We present a microscopic description of the α dependence of the voltage noise in terms of spin-current fluctuations and ISHE.