Cooling and tapping system for a gas turbine engine and corresponding operation method

An operation method for a gas turbine engine (1) and its respective cooling system including: compressing a working fluid by means of a compressor (3); feeding this compressed working fluid into at least one combustion chamber (5) where it is overheated; expanding this overheated working fluid in at least one expansion turbine (6) to produce energy; carrying out a first tapping (21,F5) of the compressed working fluid from the compressor to feed it into a first cavity (S1) of the turbine for cooling; carrying out a second tapping (23,F6) of the working fluid downstream from the first tapping to feed it into a second cavity (S2,S3) of the turbine upstream from the first cavity, for cooling; and fluidly connecting the first tapping to the second tapping to selectively feed the first tapping by means of a part of the second tapping during partial load operation conditions to keep the temperature of the first cavity within acceptable limits for the resistance of the respective turbine materials.