  $\alpha'$ corrections to near horizon dynamics of a Schwarzschild black hole
in a large number of spacetime dimensions $D$ are governed by the worldsheet
theory composed of the cigar CFT and the classical sigma model on the sphere at
the horizon, along with a timelike-Liouville theory of central charge $26-D$.
At leading order in weak string coupling, blackhole thermodynamics is
insensitive to the details of timelike Liouville theory. In this limit, we use
Lewkowycz-Maldacena-trick motivated infinitesimally off-shell closed string
worldsheet formalism in [arxiv: 2310.02313] to calculate thermal entropy
exactly in $\alpha'$. The leading term in $\alpha'\to 0$ limit and the first
stingy correction of our result is in precise agreement with the target space
Callan-Myers-Perry formula. Also, we point out a remarkable simplification at
an ultra-low temperature of order $1/\sqrt{\alpha'D}$ and show that for a
certain special set of vertex operators involving modes of the $D$ dimensional
spacetime and the time-like Liouville theory, the resulting worldsheet theory
is related to Virasoro minimal string.
