  We study the thermal partition function of Jackiw-Teitelboim (JT) gravity in
asymptotically Euclidean $AdS_2$ background using the matrix model description
recently found by Saad, Shenker and Stanford [arXiv:1903.11115]. We show that
the partition function of JT gravity is written as the expectation value of a
macroscopic loop operator in the old matrix model of 2d gravity in the
background where infinitely many couplings are turned on in a specific way.
Based on this expression we develop a very efficient method of computing the
partition function in the genus expansion as well as in the low temperature
expansion by making use of the Korteweg-de Vries constraints obeyed by the
partition function. We have computed both these expansions up to very high
orders using this method. It turns out that we can take a low temperature limit
with the ratio of the temperature and the genus counting parameter held fixed.
We find the first few orders of the expansion of the free energy in a closed
form in this scaling limit. We also study numerically the behavior of the
eigenvalue density and the Baker-Akhiezer function using the results in the
scaling limit.
