Target Wake Time (TWT), introduced in IEEE 802.11ax, lets an access point schedule exactly when each station wakes, transmits, and dozes. Existing TWT schedulers optimize energy or throughput, treating information freshness at best as a constraint and offering no performance guarantees. We design the wake schedule itself for freshness: minimize the weighted average Age of Information (AoI) over stations subject to per-station energy budgets, where the decision variables are the TWT triples (wake interval, offset, service period duration). We derive a renewal-exact AoI model for TWT under per-SP block fading and validate it against packet-level 802.11ax simulation with ~1% mean error. We show that, unlike preemptive scheduling, non-preemptive TWT packing can be infeasible at schedule density 1, and identify the granularity condition under which a small-first best-fit packer provably succeeds. Around this we build Harmonic-Greedy, a scheduler combining a convex relaxation, anchor-optimized power-of-two rounding, and a best-of-uniform safeguard, and prove it is a constant-factor approximation: 4/ln 2 ~= 5.77 under a mild granularity assumption and 6/ln 2 ~= 8.66 unconditionally. We implement the complete system in ns-3 -- a TWT wake/doze mechanism integrated with the power-save architecture, plus the scheduler -- and show that it is the only scheduler that stays near a relaxation lower bound across all regimes: against a strong energy-greedy baseline it ties when per-station energy floors already pin the periods, and wins by 4-36% exactly where the schedule density is binding and must be redistributed by AoI weight or channel quality rather than by energy budget -- the regime our analysis identifies.

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