  We provide general results on the Hagedorn temperature of planar, strongly
coupled confining gauge theories holographically dual to type II superstring
models on curved backgrounds with Ramond-Ramond and Kalb-Ramond fluxes and
non-trivial dilaton. For exact backgrounds the Hagedorn temperature is
determined up to next-to-next-to-next-to-leading order (NNNLO) in an expansion
in $\alpha'$; in all the other cases the results can be safely trusted up to
NNLO. To reach these goals we exploit two complementary approaches. On the one
hand, we perform an extrapolation to the Hagedorn regime of world-sheet results
obtained from the semiclassical quantization of string configurations winding
around the compact Euclidean time direction. En passant, we provide a detailed
derivation of the fermionic part of the world-sheet spectrum, which is hard to
find in the literature. On the other hand, we perturbatively solve the
equations of motion for the thermal scalar field corresponding to the lightest
mode of the winding string, which in flat space becomes tachyonic above the
Hagedorn temperature. The interplay between different approaches is surely
convenient, but we provide insights about a possible derivation of the whole
NNLO correction to the Hagedorn temperature from a pure world-sheet
perspective; furthermore, we determine the effective mass of the thermal scalar
from the world-sheet in full generality.
