 Solution

Varying the boundary period and filling it by the corresponding smooth Euclidean saddle computes the same canonical partition function. Because each bulk metric obeys the Einstein equation, the implicit first-order metric variation of the on-shell action reduces to boundary terms; regularity relates the varied horizon radius to the varied period. The thermodynamic identitythen yields the same Bekenstein-Hawking entropy. The conical method is an off-shell way to isolate the local horizon term, while the smooth-saddle method packages that term into the variation of the entire solution.