<p>We propose boundary terms for the action of the NS sector of supergravity on <i>M</i>₃ × <i>S</i>³ × <i>T</i>⁴ spacetimes — where <i>M</i>₃ is an AdS₃ or a linear dilaton background — that render the Brown-York stress tensor and the on-shell action finite. For AdS₃ backgrounds, we show that the on-shell action yields a free energy with chemical potentials determined by the <i>B</i>-field. TsT (T-duality + shift + T-duality) transformations of these backgrounds generate classes of linear dilaton backgrounds distinguished by their boundary conditions. Among them, there is one class of backgrounds that reproduces features of the single-trace <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <mi>T</mi> <mover accent="true"> <mi>T</mi> <mo stretchy="true">¯</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">\( T\overline{T} \)</EquationSource> </InlineEquation> deformation. These backgrounds have additional chemical potentials that can be turned off by large gauge transformations of the <i>B</i>-field. We show that the Brown-York stress tensor of these backgrounds reproduces the energies and the trace flow equation of <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math display="inline"> <mi>T</mi> <mover accent="true"> <mi>T</mi> <mo stretchy="true">¯</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">\( T\overline{T} \)</EquationSource> </InlineEquation>-deformed CFTs. We also show that the on-shell action matches the <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math display="inline"> <mi>T</mi> <mover accent="true"> <mi>T</mi> <mo stretchy="true">¯</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">\( T\overline{T} \)</EquationSource> </InlineEquation> partition function and discuss the interpretation of these results.</p>