Effect of temperature on radioactivity?

What do you mean by "prove"? If you mean in a strict mathematical sense, then looking for such assurances is a lost cause.

There are quite a wide variety of papers on this matter. Curie attacked this particular problem in 1913 with radium. They immersed a radium source in liquid hydrogen for more than an hour and didn't find a change of more than 0.1% in its activity. You can read more from the paper by Curie & Kamerlingh Onnes entitled, "The radiation of radium at the temperature of liquid hydrogen" in KNAW, Proceedings, 15 II, 1912-1913, pp. 1430-1441. People even claimed, from Russia, that polonium's activity varied depending on geography. Hardly the case.

More recently, work has been done on the half-life decay rate of $^{97}{\rm Ru}$ without seeing a noticeable temperature dependence near 20K compared with RT. See the paper by Goodwin, Golovko, Iacob and Hardy entitled, "Half-life of the electron-capture decay of $^{97}\rm Ru$: Precision measurement shows no temperature dependence" in Physical Review C (2009), 80, 045501, arXiv:0910.4338.

It could be that there is a small dependence, but not even the Russian paper mentioned above by Martin agrees there is a measurable temperature dependence.