I'm playing a Hexblade with Pact of the Blade, and I was wondering whether the weapon you use as part of the Hex Warrior ability counts as an arcane focus, or only if it's a pact weapon and you have Improved Pact Weapons Invocation?
A Hex Weapon isn't an arcane focus automatically
The Hex Warrior class feature (XGtE, p. 55) says:
Whenever you finish a long rest, you can touch one weapon that you are proficient with and that lacks the two-handed property. When you attack with that weapon, you can use your Charisma modifier, instead of Strength or Dexterity, for the attack and damage rolls. This benefit lasts until you finish a long rest. If you later gain the Pact of the Blade feature, this benefit extends to every pact weapon you conjure with that feature, no matter the weapon's type.
Nowhere does it mention anything about the weapon being an arcane focus, so it isn't one.
As you've noted, the Improved Pact Weapon eldritch invocation (XGtE, p. 57) can allow you to use your weapon as an arcane focus, since it says:
You can use any weapon you summon with your Pact of the Blade feature as a spellcasting focus for your warlock spells.
So you would need this eldritch invocation to use your hex weapon as an arcane focus. However, your hex weapon would also need to be your pact weapon, since it says "with your Pact of the Blade feature", so you would need to choose your hex weapon as the magic weapon you shunt out of existence and resummon with Pact of the Blade, as per the PHB (p. 107):
You can transform one magic weapon into your pact weapon [...] You can then dismiss the weapon, shunting it into an extradimensional space, and it appears whenever you create your pact weapon thereafter.
If you were to choose your hex weapon as your pact weapon, then it would count as a pact weapon to satisfy the Improved Pact Weapon eldritch invocation. At that point, your hex weapon could be used as an arcane focus.