It's a misconception brought on by restored aircraft. The interior cowls on P-47s were not painted. With a radial engine slinging oil all about, what would be the point?
Here's a nice image of a P-47 razor-back. Luckily, the sun is shining on the interior of the cowl. It's not painted white, but is kind of an anodized aluminum (about the same color as that drop tank).
Hope this helps.
PS> Naturally, this subject has come up before, so I found an old post with the IPMS Stockholm information on P-47 colors.
Republic P-47 Thunderbolt
The subject of cockpit colours of the P-47 seems to have thus far defied conclusive analysis. Surviving P-47s and contemporary photos show a dark green shade in the cockpit, similar or possibly equal to Dull Dark Green. This is in contrast with the available Erection and Maintenance manuals which invariably call for green-tinted primer in cockpit areas.
The 1944 Erection and Maintenance Instructions covering P-47C, G and D state that "cockpits shall be finished with one coat of tinted zinc chromate primer to eliminate glare resulting from untinted primer." As can bee seen, the use of "tinted primer" is not consistent with the Dark Dull Green found in other evidence.
Perhaps an explanation is to be found in the formula of tinted primer given in the above manual. Nowhere in the above document is the tinted Zinc Chromate specified to match ANA Interior Green. Instead, the specifications include a rudimentary mixing formula, described as one gallon Black to one gallon Yellow Zinc Chromate primer. The formula is probably an error. If the intended colour was to be Interior Green, the document should have stated 1/10 gallon Black to 1 gallon Zinc Chromate, consistent with other Erection and Maintenance documents of the period.
A possibility remains that Republic followed the instructions to the letter, obtaining some sort of black-green colour for the cockpit areas. Other hypotheses claim that the colour used could be Bronze Green or Dull Dark Green. Another mystery.
Another conventional wisdom states that Curtiss-built P-47Gs differed from Republic-build P-47Ds by having Interior Green (actually, Curtiss Cockpit Green) in the cockpit and wheel well areas. However, this does not seem to be consistent with examination of wrecked P-47G parts, which show Dark Dull Green in the cockpit.
Since there were less than 200 P-47Gs made and they were only used for training in the US, this controversy is of limited interest to modellers, which would usually be interested in Republic-made Thunderbolts.
According to the Erection and Maintenance manuals, the fuselage decking under the bubble canopy of the P-47D from the windscreen to the area aft of the cockpit armour plating, was to be painted Dark Olive Drab 41, the same colour being specified for the anti-glare area of the forward fuselage. Armour plating was specified to the same colour as the interior finish of the cockpit.
Another yet unresolved mystery is the turtleback area beneath the rearmost cockpit window of the razorback versions. Many variants have been called for, but the most likely choices (based on the available contemporary colour photographs) are Olive Drab for the early camouflaged aircraft, and some kind of medium grey further down in the production.
According to factory instructions, the fuselage decking inside the canopy on bubbletop Thunderbolts was to be painted in Olive Drab, with the inside of the canopy framing in flat black. The rear armour plate in the cockpit was to be painted to match the cockpit interior colour.
Interiors of P47 aircraft cowlings were natural metal. The aluminium in this area was anodised giving a darker and very dull greyish appearance. The engine firewall was left unpainted. Engine mounts were primed in Zinc Chromate Green.
All other interior surfaces of the fuselage with exception of the firewall were finished in Zinc Chromate Yellow. This included also wheel wells, undercarriage covers and armament compartments in the wings.
Undercarriage legs were painted Dark Olive Drab 41 on camouflaged aircraft. This practice continued over to at least some natural metal machines. At some point in production the requirement seems to have changed to allow an Aluminium lacquer finish to be used.