I am not sure why we don't have any terminal nodes updates for Cox-Ph model. Please find the note of gbm package in R showing the formulas for updating the terminal nodes. It uses Newton-raphson single step update like the classification model.
Paper Link - https://cran.r-project.org/web/packages/gbm/vignettes/gbm.pdf
Code Link - https://github.com/harrysouthworth/gbm/blob/master/src/coxph.cpp
I am not sure why we don't have any terminal nodes updates for Cox-Ph model. Please find the note of gbm package in R showing the formulas for updating the terminal nodes. It uses Newton-raphson single step update like the classification model.
Paper Link - https://cran.r-project.org/web/packages/gbm/vignettes/gbm.pdf
Code Link - https://github.com/harrysouthworth/gbm/blob/master/src/coxph.cpp