Spontaneous decay rate are considered to be a kind of intrinsic property and it should not depend on anything else, see the answers in the related question. Usually when there are particles interacting with the atom, the decay rate can change but it is not spontaneous. I want to add an experimental results.
There are occasional reports of the observing decay rate fluctuate less than 1%. A recent experimental results claimed that the effect of the sun on the beta decay can be large and its variation depends on the hour of day and the date of year. Details and discussion are given in a blog post (Original paper).
They have record the beta decay of $^{214}Pb$ and $^{214}Bi$ with 28733 measurements of gamma radiation. The time series is shown in Figure 1 below. The power of the signal over days and years are shown in Figure 2 and it is very close to the power of sun in Figure 3.
The data here is pretty clearly show that there is large fluctuation overtime and there is a relation to the sun. Though, it is hard to tell whether the spontaneous decay rate has been changed. But I think it might be some unknown interaction mechanism with particles from sun.
Figure: Gamma measurements, normalized to mean value unity, as a function of time.
Figure: Gamma measurements as a function of date and time of day. The color-bar gives the power, $S$, of the observed signal.
Figure: Solar elevation as a function of date and time of day. The color-bar gives the power, $S$, of the observed signal.
Edit: I want to add a note about the discussion of the previous question, and pointed by @dmckee that this result and others are conducted by the same author. That is these are not independent results. It is therefore less trustable.
However, the results produced by their group this time is much clear and stronger. I think there are other research groups interested to reproduce it. This experiment should be much easier than the previous one for other groups to valid its correctness (or figure out stupid experimental problem). Hence, it could be expected that we can see more independent results to verify it soon.