Ever since the approval system was overhauled in Beta Four, Approval Relics have been a little wonky.  In Beta 4, they didn't work at all.  In Beta 5, they work, but are still buggy.  They don't give the full approval bonus to colonies that planets are supposed to get, I don't know what the precise formula is, but whatever it was, it wasn't 10% per module installed as it was supposed to be.  Worse, when a player goes over 100 planets, their effectiveness plummets.  I had already submitted a couple of tickets on this and started/commented a few threads on this problem.  In those threads, androshalforc suggested that the XML for Approval Relics was to blame.  Well, I finally broke down and tested it myself.

Yep.  That was it.  At least, now that I have modified the entry for Approval Relics in the file StrategicResourceDefs.xml, I can see the approval bonus on the planet screen for the first time since Beta 3:

I tested out the math, and it checks out (within rounding, at least)

I haven't seen what happens as I pass the 100 planet "barrier", but I presume it should work fine.

What I did was create a simple mod that has the following change in the file StrategicResourceDefs.xml.  

I changed the following entry in that file from 

<StrategicResource>
<InternalName>ApprovalRelic</InternalName>
<DisplayName>APPROVAL_RELIC_LABEL</DisplayName>
<Type>Relic</Type>
<ArtDefine>MoraleRelic</ArtDefine>
<HowToTapText>STRATEGIC_RELIC_TOOLTIP_UNTAPPED_MSG</HowToTapText>
<Resource>
<EffectType>Approval</EffectType>
<Scope>Global</Scope>
<Target>
<TargetType>Colony</TargetType>
</Target>
<BonusType>Multiplier</BonusType>
<ValueType>Special</ValueType>
<SpecialValue>
<Special>GetValueFromStarbase</Special>
</SpecialValue>
</Resource>
<Resource>
<EffectType>Research</EffectType>
<Scope>Global</Scope>
<Target>
<TargetType>Colony</TargetType>
<TargetQualifier>IsAncient</TargetQualifier>
</Target>
<BonusType>Multiplier</BonusType>
<ValueType>Special</ValueType>
<SpecialValue>
<Special>GetValueFromStarbase</Special>
</SpecialValue>
</Resource>
<Prerequ>
<StarbaseModule>XenoArcheologyLabModule</StarbaseModule>
</Prerequ>
</StrategicResource>

to

<StrategicResource>
<InternalName>ApprovalRelic</InternalName>
<DisplayName>APPROVAL_RELIC_LABEL</DisplayName>
<Type>Relic</Type>
<ArtDefine>MoraleRelic</ArtDefine>
<HowToTapText>STRATEGIC_RELIC_TOOLTIP_UNTAPPED_MSG</HowToTapText>
<Resource>
<EffectType>GoodsAndServices</EffectType>
<Scope>Global</Scope>
<Target>
<TargetType>Colony</TargetType>
</Target>
<BonusType>Multiplier</BonusType>
<ValueType>Special</ValueType>
<SpecialValue>
<Special>GetValueFromStarbase</Special>
</SpecialValue>
</Resource>
<Resource>
<EffectType>Research</EffectType>
<Scope>Global</Scope>
<Target>
<TargetType>Colony</TargetType>
<TargetQualifier>IsAncient</TargetQualifier>
</Target>
<BonusType>Multiplier</BonusType>
<ValueType>Special</ValueType>
<SpecialValue>
<Special>GetValueFromStarbase</Special>
</SpecialValue>
</Resource>
<Prerequ>
<StarbaseModule>XenoArcheologyLabModule</StarbaseModule>
</Prerequ>
</StrategicResource>

I did this via a simple one file mod so I wouldn't muck up my main game.  I will play a game with this change, but I strongly suspect that this is the culprit.  As a happy coincidence it should make playing on maps with lots of planets a better experience.  Maybe not a perfect experience, as the faction size penalty still needs looking at IMO.  But at least this should help.