$T^2$ deformations in the double-scaled SYK model: Stretched horizon thermodynamics

We study T$^2$-deformations in the DSSYK model \emph{after} performing ensemble averaging, to learn about the properties of its bulk dual geometry according to finite cutoff holography. We first derive the flow equation for generic holographic quantum mechanical systems dual to dilaton gravity theories without requiring AdS asymptotic boundaries, resulting in a new dictionary entry relating the boundary deformation parameter to the bulk radial boundary cutoff. Applying the results to the DSSYK model, we show that the T$^2$ deformation can probe regions of positive and approximately constant curvature in sine dilaton gravity. The dS$_2$ stretched horizon holographic proposal by Susskind \cite{Susskind:2021esx} is recovered as the radial cutoff approaches the horizons of the bulk dual. The thermodynamic quantities of the system exhibit enhanced growth in this limit. In the deformed boundary theory, we uncover phase transitions from thermodynamically stable to unstable configurations depending on its temperature, which also scrambles information at a (hyper-)fast rate in terms of its out-of-time-ordered correlator.