What temperature is required to affect stability of the nucleus According to answers to "What is the temperature of an atomic nucleus?" the temperature inside a nucleus is in the thermal equilibrium with outside environment.
So for an outside temperature of 10^N, for N=2 the effect is virtually zero, but at N=12 the nucleons will be turned into Quark-Gluon plasma. At some smaller N, I suppose, the nucleons will be intact but the nucleus will be unbound (turned into baryon plasma/hydrogens?).
What is the N (and therefore temperature) that would noticeably affect the nucleus, such as decreasing the half-life of Radium nucleus by one second?
 A: If you poke around in the Evaluated Nuclear Structure Data File, you might be able to find some nuclei with excited states whose strong or weak decay probabilities are greater than the corresponding decays of the ground state. There won't be a lot of data, because usually electromagnetic transitions are fast enough that observing the particle decays from the excited states is just impossible.  But there is some data from isomeric states.  For example, hafnium-182 has a low-energy, high-spin isomer with a beta decay partial width of hours, while the ground state has a beta decay lifetime of megayears.
Once you've identified some candidate states, you can compute the relative probability of finding those states occupied, relative to the ground state, using the Boltzmann factors,
$$
\frac{P(E)}{P(E_0)} = \exp\frac{E_0 - E}{kT}
$$
Since nuclear states generally have excitation energies of mega-eV above the ground state, you need $kT\sim\text{mega-eV}$ before the excited states have non-negligible thermal occupation.
