I have built a NK model with separable utility function, without capital and a production function with constant return to scale.
I have assumed labour to be 0.33 in steady state and included Frisch Elasticity as an endogenous variable.
This results in a negative Frisch elasticity in steady state.

Is it normal in this kind of environment?
Obviously, it does not have an economic sense.

Do you think that I would get a more sensible value once capital is into the model?

The Frisch elasticity is still a parameter. I don’t understand what you are doing. When you calibrate labor in steady state, the disutility parameter needs to be adjusted accordingly. But that is different from the Frisch elasticity. See e.g.

which features a unit Frisch elasticity, but sets the labor disutility parameter phi to have labor as 1/3

I have tested your code RBC_baseline by building an alternative version in which the parameter psi is defined as an endogenous variable - see attached.
This alternative way of coding the labour disutility parameter to hit labour in steady state seems to lead exactly to the same results. However, there is an advantage as one can see the value of the parameter in steady state and this would not be possible by defining the psi in the parameter block.

This is from the endogenous variable version:
STEADY-STATE RESULTS:

y 1.04578
c 0.571206
k 10.8761
l 0.33
z 0
ghat 0
r 0.126923
w 2.12325
invest 0.261445
log_y 0.0447641
log_k 2.38657
log_c -0.560006
log_l -1.10866
log_w 0.752949
log_invest -1.34153 psi 2.49049

Going back to my initial question. In the NK model, I extrapolate the labour disutility parameter by combining the wage supply and the wage demand. Unfortunately, this leads to:
phi_par=log(H)/log(PWPA^(1-gamma)(Disp*CY)^(-gamma))-gamma;
which, then results in a negative value for the parameter.
Do you think this to have economic sense?

I have solved the issue of the negative Frisch elasticity in steady state by means of a “trick”.
Practically, I have modified my utility function by adding a scaling factor to labour disutility:

U= F( C ) - PSI *(1/(1+PHI)*L^(1+PHI)

In this way, It is possible let PSI adjusting to hit given values for both labour in steady state and the inverse of the Frisch elasticity, PHI.

What is your opinion on this strategy?
I have seen some papers doing this.

What would be the interpretation of this scaling parameter?

I am not sure I am following. The new felicity function is the standard one. PSI measures the weight of leisure relative to consumption in your felicity function. Before, you simply set the weight to 1, but there is not rationale for doing that.