There is NO violation of the 1st law

To be honest with you I had at the beginning a serious dilemma:

Extra energy due to spin alone contravenes the first law.

It took me many sleepless nights to figure out what exactly happens here.

There is not any extra energy involved. There is not any additional energy "produced" by planet's faster spinning.

The energy from solar flux in, equals the energy reflected and emitted by the planet's surface out to space.

To explain the phenomenon of a planet being warmer when spinning faster one should refer to the Stefan-Boltzmann Law non-linearity.

I have a paradigm of how it happens here:

The faster a planet rotates ( n2 > n1 ) the higher is the planet’s average (mean) temperature T↑ mean

T min ↑↑ → T mean ← T↓ max

when n2 > n1

( it happens because T min ↑ grows faster than T↓ max goes down )

It happens in accordance to the Stefan-Boltzmann Law.

Let’s explain: Assuming a planet rotates faster and Tmax1 - Tmax2 = 1 oC.

Then, according to the Stefan-Boltzmann Law:

Tmin2 - Tmin1 > 1oC

Consequently

Tmean2 > Tmean1

Assuming n2 > n1 the solar irradiated hemisphere average temperature T1 - T2 = 1oC

Then the dark hemisphere average temperature

T2 - T1 > 1oC

Consequently the total average

Tmean2 > Tmean1

So we shall have: when n2 > n1

T min ↑ → T↑ mean ← T↓ max

The faster a planet rotates ( n2 > n1 ) the higher is the planet’s average (mean) temperature T↑ mean.

When a planet rotates slowly the solar irradiated hemisphere warms at higher temperature. Consequently a warmer surface emits

Jemitt. = σΤ

When a planet rotates faster the solar irradiated hemisphere warms at lower temperature.

Consequently a colder surface emits

Jemitt. = σΤ⁴ , and

Jemitt. > Jemitt.

In both cases, slow or fast, the rotating planet absorbs the same amount of energy:

Jabs = Φ (1-a) So (1/R²)

The difference of

Jemitt. - Jemitt.

is what keeps the faster rotating planets warmer, everything else equals.

Now we should focus on what happens at the planet's dark side. As it was said "The change would be the difference between dawn and dusk temperatures, which would be smaller with a faster rotation period".

At dusk a faster rotating planet will have a higher local temperature.

At dawn a faster rotating planet would have a higher local temperature.

The new day for the faster rotating planet starts with a warmer surface.

At the culmination hours in the midday the slow rotating planet surface warms much higher and emits much more energy out to space

Jemitt. = σΤ⁴ compared with

Jemitt. = σΤ⁴ .

Τ > Τ

and due to the Stefan-Boltzmann Law non-linearity we have

Τ⁴ >>> Τ

so we have Jemitt. >>> Jemitt.