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We have measured the temperature dependence of the low-frequency noise properties across for mesa stacks of intrinsic Josephson junctions (IJJs) in single crystals of Bi2Sr2CaCu2Oy (BSCCO) in the absence of an external magnetic field. The measured curves for noise spectrum density SV(f) showed a 1/f frequency dependence at temperatures in the range from 4.3 K to 150 K except 36 K. We estimated the magnitude of the SV(f) by using Rogers and Buhrman's empirical theory and Machlup's formula. From the observed curves for SV(f), we estimated a 1/f noise level η for a BSCCO IJJ biased at a sub-gap voltage of approximately 10-5 μm2, and this value was almost completely independent of temperature in the range between 4.3 K and 40 K. We also found that the η values above Tc were smaller by a factor of approximately 10 than those below Tc. The η value for the BSCCO IJJ is approximately two orders of magnitude greater than for an epitaxial NbN/MgO/NbN tunnel junction biased above-gap voltage. We consider that the observed variation in the noise level with temperature is not interpretable in terms of a magnetic origin and the 1/f noise in a BSCCO IJJ may be somehow associated with stress or defects in the BiO and/or SrO layers act as tunnel barriers.