Before revision:  We examine the main thermodynamic consequences of the existence of this type of multifractality related to the thermal stable, metastable, and unstable phases within a hierarchy of fluctuations, and also to the first and second order phase transitions between them  .
After revision:  We examine the main thermodynamic consequences of the existence of this type of multifractality . They can be expressed directly in the language of thermally stable, metastable, and unstable phases within a hierarchy of fluctuations, and also to the first and second order phase transitions between them .