Kagome antiferromagnets are predicted to host exotic field-induced phases, yet experimental identification of their thermodynamic signatures remains challenging. Here we combine specific heat (C) and ultrasound measurements up to 36 T and down to 0.5 K to map the H-T phase diagram of the anisotropic kagome magnet Y₃Cu₉(OH)₁₉Cl₈. We show that the suppression of long-range antiferromagnetic order with field at ∼ 15 T approximately coincides with the onset of the unconventional 1/6 (or 1/9) magnetization plateau previously observed in this system. The attenuation (and sound velocity) then intriguingly increases down to 0.5 K just at μ₀H = 15 T, and C/T becomes temperature independent. Moreover, our measurements are consistent with the opening of a spin gap Δ₀≃3.5± 0.3 K within the plateau and reveal a further compressible (ungapped) phase beyond the plateau phase, unveiling a rich sequence of field-induced phases in this quantum magnet.