Hospital

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After each battle, your troops will come home weary of their wounds and will slowly regain their strength with time. If you wish to give time a gentle push, let them be healed by putting doctors in the Hospital. You also may spend your gold on expensive healing balms that restore your troops regardless of whether they are in the Hospital or not.

Hospital increases capital health regeneration rate, or Healing Rate, and heals instantly for gold.

Building Hospital requires Garrison

Hospital unlocks the "Heal" button which allows you to spend gold to heal vassals instantly; 10 gold per point.

[edit] Healing

Hospital production accelerates all vassals' healing for entire capital.
Hospital upgrades increase the healing rate for capital.

Healing time is ~5 hours from zero to full health (not confirmed) without anyone in the hospital. Therefore, the healing "rate" will vary by vassal, similar to Church zeal rates. e.g. if a 20 takes 4 hours to heal from 0 to 20, the rate is 1 hit-point per 12 minutes. A 10 would therefore have a heal rate of 1 point per 24 minutes. You can elect to "auto-heal" your vassals, at the rate of 10 gold per hit-point, instead of waiting for healing. Populating the hospital speeds the healing for all vassals in the kingdom.

Hospital Healing Rate = Base Healing Rate * (1 + [Healing Bonus % / 100])

where

Healing Bonus % = SUM(vassal power in hospital) x [1 + (# upgrades x 0.05)] x 300 / (total vassal power excluding serfs/steward)

100% means your vassals heal in half the regular time. Per the release notes, the healing bonus displayed in the hospital is proportionate to hospital production divided by total vassal power.