when using temporal breach on thousand tail watcher. does it play it free including acceleration cost?

No — Temporal Breach makes Thousand-Tailed Watcher free, but the Accelerate cost is not included; you can still choose to pay it.

"Ignoring its cost" only zeroes the base costs (7 energy, 1 mind power). Accelerate is an optional additional cost (Rule 805.2), so you may pay 1 energy + 1 mind power to have the Watcher enter ready, or decline and let it enter exhausted (Rule 356.1.b.1, 356.2.b). This matches the FAQ rulings for Baited Hook, Portal Rescue, and Z-drive — "ignore costs" never waives optional additional costs like Accelerate (FAQ #10924, #7891).