Gauge invariant perturbations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow>

Abstract The Gauss–Bonnet invariant connects foundational aspects of geometry with physical phenomena in a variety of ways. Teleparallel gravity (TG) offers a novel direction in which to use the Gauss–Bonnet invariant to go beyond standard cosmology. In this work, we explore the cosmological perturbations of TG generalized through the Gauss–Bonnet invariant. This is crucial in understanding the viability of these models beyond background analyses. We do this by taking a gauge-invariant approach, followed by popular gauge choices. It is essential to take this approach to understand the stability and healthiness of the underlying theory. We determine the equations of motion for all perturbative modes and provide a physical interpretation of the new contributions for each mode.