The concept of nuclear spin temperature has been a cornerstone of the theory of dynamic nuclear spin polarization by electrons in various semiconductor structures for decades. Still, it is not always applicable to strongly localized electrons due to their long spin correlation times. This motivated the use of the oversimplified central spin model for the description of the nuclear spin dynamics in quantum dots. Here, we present a microscopic theory that bridges the gap between these two approaches by describing the nuclear spin thermodynamics for systems with long electron spin correlation times. Importantly, our theory predicts that efficient nuclear spin cooling by strongly localized electrons requires an external magnetic field by far exceeding the local field of nuclear spin-spin interaction, and that the time of the nuclear spin heating by unpolarized electrons may change by several orders of magnitude depending on the magnetic field.