What does the Pax Romana casus belli cost?

piety and prestige; the amount depends on the titles, claims and distance, worked out when you declare.

Casus belli · Invasions

Pax Romana

The war is called “Roman Invasion of …”

Pax Romana is a casus belli in Crusader Kings III (invasions). If the attacker wins: The attacker seizes all titles within the defender's realm belonging De Jure to the target title. If the defender wins: The defender keeps their titles. Declaring it costs piety and prestige. The war goal is worth 80% war score. Patch 1.20.0.3, from game/common/casus_belli_types/00_invasion_war.txt.

Quick answers (3)

What do you get for winning a Pax Romana war in CK3?: The attacker seizes all titles within the defender's realm belonging De Jure to the target title (patch 1.20.0.3).
What happens on a white peace?: The defender keeps their titles.
What does the Pax Romana casus belli cost?: piety and prestige; the amount depends on the titles, claims and distance, worked out when you declare.

How the war ends

Attacker wins

The attacker seizes all titles within the defender's realm belonging De Jure to the target title.

White peace

The defender keeps their titles.

Defender wins

The defender keeps their titles.

If the war becomes invalid

Roman Invasion of … invalidated!

Who can declare it

The game shows no text of its own for these conditions; the script is below.

Every invasions war (the group's allowed_for_character) (7 lines of game script)

herders_and_tributary_constraints = yes
pam_personal_pacifism_offensive_war_ban_trigger = yes
tgp_dynastic_cycle_offensive_wars_ban_trigger = yes
tgp_war_ban_all_ministers_trigger = yes
NOT = {
	government_has_flag = government_is_theocracy
}

As written in game/common/casus_belli_groups/00_casus_belli_groups.txt.

Attacker (allowed_for_character) (6 lines of game script)

is_roman_emperor_excluding_byzantium_trigger = yes
government_has_flag = government_is_administrative
any_owned_story = {
	type = ep3_story_cycle_restoring_rome
}
prestige_level >= 1

As written in game/common/casus_belli_types/00_invasion_war.txt.

Defender (allowed_against_character) (3 lines of game script)

scope:attacker = {
	top_liege != scope:defender.top_liege
}

As written in game/common/casus_belli_types/00_invasion_war.txt.

When it can start (valid_to_start) (3 lines of game script)

scope:target = {
	tier = tier_empire
}

As written in game/common/casus_belli_types/00_invasion_war.txt.

Cost to declare

Costs piety and prestige. The amount is worked out when you declare (by the titles targeted, claims, distance and more), so the script is shown as written:

Cost (11 lines of game script)

piety = {
	value = 0
	add = common_cb_impious_piety_cost
}
prestige = {
	add = {
		value = 200
		desc = CB_BASE_COST
	}
	multiply = common_cb_prestige_cost_multiplier
}

As written in game/common/casus_belli_types/00_invasion_war.txt.

Source: game/common/casus_belli_types/00_invasion_war.txt and the English localisation, Crusader Kings III 1.20.0.3. “The attacker”, “the defender” stand where the game names the character; “…” marks other words it fills in during play.