Round Down
Some players and GMs may prefer replacing this rule...
ROUND APPROPRIATELY
Follow standard rules for rounding; if a number ends in one-half or greater, round up, while if a number ends in less than one-half, round down.
...with this rule:
ROUND DOWN
There’s one more general rule you need to know at the outset. Whenever you divide a number in the game, round down if you end up with a fraction, even if the fraction is one-half or greater, unless the feature says otherwise.
This variant rule used to be the official rule, and when it was, several features in the game had exceptions to it. These were typically written like this, "Add half your proficiency bonus (rounded up), to the result", or this, "Equal to half your Intelligence modifier (round up) to the result."
Most of these exceptions aren't present in the current rules. In order to use this variant rule as it originally was used, the following features involving fractions should be rounded up; unaffected by this variant rule. (GMs should consider applying this exception to other features as well, where they deem appropriate.)
Player's Handbook
Class Features and Hit Dice
The average value of a Hit Dice optionally added to maximum hit points when leveling up, equal to half the hit die's maximum value. (The same applies to a ship's Hit Dice when increasing its tier.)
Brute Force (Assault Specialist)
The increase to Strength, Dexterity or Constitution checks, equal to half your proficiency bonus.
Techcaster Archetypes
The tech points it has, which includes half the relevant class's level. (Shield Specialist, Industrial Approach, Rakatan Order, Saboteur Practice, Slicer Pursuit, Mechanist Technique.)
Vow of Serenity (Monk)
The increase to maximum focus by an amount equal to half your Wisdom or Charisma modifier.
Ideal of the Tranquil (Sentinel)
Both the temporary and normal force points equal to half your Wisdom or Charisma modifier.
Phasethrow (Path of the Forceblade)
The additional damage equal to half your Wisdom or Charisma modifier.
Heavy (Weapon Properties)
The additional damage equal to half your Strength modifier.
Defense Style (Fighting Styles)
The reduction to damage taken equal to half your Strength or Constitution modifier.
Equilibrium Style (Fighting Styles)
The increase to AC and Dexterity saving throws equal to half your Dexterity modifier.
Wretched Hives
Crafting and Blueprints
The amount of blueprints that can be learned, equal to half your character level.
Membership in Multiple Factions
The maximum of total combined ranks across all factions, equal to half your character level.
Class Enhanced Items
The number of charges it can hold, equal to half the relevant class's level. (Boots of Bounding, Gauntlets of the Warrior, Circlet of Persistence, Goggles of the Tinkerer, Necklace of Determination, Cape of the Orator, Cowl of the Professional, Mantle of the Anomic, Pauldron of Retribution, Vambraces of the Outrider.)
Ballistic Polymer
Damage reduction equal to half your proficiency bonus.
Burst Core and Rapid Core (All Rarities)
The burst number equal to a fraction of the reload number.
Expanded Content
Empowered Connection (Way of Confluence)
The temporary hit points equal to half your Wisdom or Charisma modifier.
Artful Dancer (Performance Practice)
The increase to AC equal to half your Charisma modifier.
Dance of Death (Performance Practice)
When halving your Sneak Attack dice to damage multiple creatures.
Beast Companion (Zoologist Pursuit)
The bonus to the companion's ability checks, armor class, attack rolls, damage rolls, and saving throws equal to half your Intelligence modifier.
Techcaster Archetypes
(Same as in the PHB.)
One More Change
Under Damage Resistancs and Vulnerability in the PHB, change "If a creature or an object has resistance to a damage type, damage of that type is reduced by half against it." to "damage of that type is halved against it." This way, damage resistance produces the same results under the variant rule as it does without it.