Depending on what you mean by "a long time," the answer is not really. Test cypionate was kept at 80 C for 48 hours and no degradation was observed when quantified by HPLC. Elevated temperature may accelerate oxidation kinetics, but in a solution of 5% hydrogen peroxide, no oxidation was observed over several hours.... this is far more oxidizing than oxic headspace within a vial. That means neither the testosterone nor the ester prosthetic group was damaged in either of those cases.
As far as it making a shorter-acting ester, that's now how that would work. The ester linkage, where the cypionate connects to the testosterone, is much less stable than the bonds linking the carbon backbone of the cypionate.
EDIT -
Here's the citation. This doesn't mean it's a good idea to store it other than the way people usually do -- cool and dark. You don't know the excipients or impurities in your batch, and this may alter the chemistry in unpredictable ways.