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"...It uses your base attack bonus (possibly allowing it multiple attacks per round in subsequent rounds) plus your Wisdom modifier as its attack bonus. It strikes as a spell, not as a weapon, so for example, it can damage creatures that have damage reduction. As a force effect, it can strike incorporeal creatures without the reduction in damage associated with incorporeality. The weapon always strikes from your direction. It does not get a flanking bonus or help a combatant get one. Your feats or combat actions do not affect the weapon...."
-- Spells and supernatural abilities (e.g., Inspire Courage) apparently do, however.
So, does Haste (or Boots of Speed, et al) grant it an extra attack?
(My assumption is no, given that the Spiritual Weapon isn't a targetable creature; but thought I'd ask anyway since I observe no instance of the question being asked previously in the messagebase.)
The spell description for haste gives the answer. Emphasis mine. Since the spell always refers to the part I've bolded, I have no idea why spiritual weapon would be affected.
The transmuted creatures move and act more quickly than normal. This extra speed has several effects.When making a full attack action, a hasted creature may make one extra attack with one natural or manufactured weapon. The attack is made using the creature's full base attack bonus, plus any modifiers appropriate to the situation. (This effect is not cumulative with similar effects, such as that provided by a speed weapon, nor does it actually grant an extra action, so you can't use it to cast a second spell or otherwise take an extra action in the round.)
A hasted creature gains a +1 bonus on attack rolls and a +1 dodge bonus to AC and Reflex saves. Any condition that makes you lose your Dexterity bonus to Armor Class (if any) also makes you lose dodge bonuses.
All of the hasted creature's modes of movement (including land movement, burrow, climb, fly, and swim) increase by 30 feet, to a maximum of twice the subject's normal speed using that form of movement. This increase counts as an enhancement bonus, and it affects the creature's jumping distance as normal for increased speed. Multiple haste effects don't stack. Haste dispels and counters slow.
It's a force effect, as stated in the spell description, which attacks on its own. It's not a creature, so couldn't be targeted by Haste or wear Boots of Speed, I believe.
But the spiritual weapon takes the form of your deity's favored weapon, so clerics of Irori can have a big spectral foot flying around and kicking people, so why not put a boot of speed on it?
Because Irori only wears slippers! And he'd probably consider using a magical boot a shortcut to perfection! (Kidding too.)