A common form of superhydrophobic surface is made out of a periodically groovedsolid substrate, wherein cylindrical bubbles are trapped in a Cassie state. When a macroscopic tem-perature gradient is externally applied, Marangoni forces generate thermocapillary flow of character-istic magnitudeU=−aGσT/μ, in which 2ais the groove width,Gthe applied-gradient magnitude,μthe liquid viscosity, andσTthe derivative of interfacial-tension coefficient with respect to the temper-ature. We consider the case of a gradient which is applied parallel to the grooves. Assuming a highlyconducting solid substrate, we seek to calculate the longitudinal velocity component, anti-parallelto the applied gradient, and in particular its “slip” value, attained at large distances away from thesurface. Normalized byU, this value depends only upon the bubble protrusion angle and the solidfraction . In this paper we consider the small solid-fraction limit 1 and focus upon the case of90◦protrusion angle, for which this limit is known to be highly singular in the comparable problemof shear-driven flow [Schnitzer, Phys. Rev. Fluids, 1 (2016), 052101]. Using matched asymptoticexpansions, we find that the dimensionless slip velocity is given byπ2/√8 + ln +I−ln(8π2) +o(1).The first two terms in this expansion follow from a lubrication-type analysis of the narrow gap regionseparating two neighboring bubbles. The subsequentO(1) terms follow from asymptotic matchingwith the bubble-scale region, where the bubbles appear to be touching. The solution in that regionis obtained using conformal mapping techniques, with the constantIgiven as an explicit integral,evaluated numerically to be 2.27.