Is it possible to laser cool without using spatially-varying magnetic fields and above zero degrees?

If, for whatever reason, you don't want to use a magnetic field, you can always chirp your laser frequency.

Most ultra-cold atoms experiments starts with a Zeeman slower for a good reason : it's cheap to make and it allows to slow all of the atoms of a given beam with a thermal velocity distribution down to something where usually all of the atoms have a velocity close to $simeq 10,text{m.s}^{-1}$.

The resonance condition between an atom with an eigen-pulsation $omega_0$ moving at a velocity $v$ against a laser with a wave-number $k$ and a pulsation $omega$ is : $$ omega=omega_0-k,v(z(t))+frac{mu_B}{hbar}B(z) $$ In most situations, the spatial configuration of the magnetic field $B(z)$ is tailord so that such resonance condition is satisfied along the whole trajectory $z(t)$ of any atom. This means that, along this scheme, the cooling process is continuous : at any point $z$ any time $t$, atoms will be cooled down.

If you don't want to use a magnetic compensation $B(z)$, nothing is stopping you from changing/chirping the laser frequency $omega(t)$ itself along the trajectory $z(t)$ of an atom, such that the resonance condition reads : $$ omega(t)=omega_0-k,v(z(t)) $$ If this condition is satisfied for any $t$, only a fraction of an atomic beam can be slowed down : only those with a given initial velocity $v_0=v(z(0))$.

Chirped laser cooling as been successfully implemented at various occasions 1 but is rarely used in usual ultracold set-up since it allows to slow atoms only batch per batch (and not continuously unlike in a Zeeman slower), thus leading generally to quite low atomic densities.

1 : Laser Manipulation of Atomic Beam Velocities : Demonstration of Stopped Atoms and Velocity Reversal W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu ; Phys. Rev. Lett. 54, 996