Table[Flatten[{h} /. FindRoot[h == F[h, b, g], {b, 1, 1000}, {g, 1, 60})

ι[m_, n_] := Binomial[n, n*(1 - m)/2]*2^(-n);
f[m_, h_, b_, g_, n_] := (h*m + g/2*m^2) +
   1/(n*b)*Log[ι[m, n]];
μ[m_, h_, b_, g_, n_] :=
  Exp[b*n*f[m, h, b, g, n] + b*n*(-h + g/2)]/
   Sum[Exp[b*n*f[x, h, b, g, n] + b*n*(-h + g/2)], {x, -1 + 2/n,
     1 - 2/n, 2/n}];
moment[h_, x_, b_, g_, n_] := Sum[m^x*μ[m, h, b, g, n], {m, -1 + 2/n, 1 - 2/n, 2/n}];
var[h_, b_, g_, n_] := moment[h, 2, b, g, n] - moment[h, 1, b, g, n]^2;
cov[h_, b_, g_, n_] := moment[h, 3, b, g, n] - moment[h, 1, b, g,n]*moment[h, 2, b, g, n];
F[h_,b_,g_,n_]:= -d*b*(cov[h, b, gg, n] +
           2 var[h, b, gg, n]);
n = 100;
d = 0.9;

glist = Table[g, {g, 0.4, 1, 0.01}];
blist = Table[b, {b, 1.1, 10.1, 0.01}];

heatdata = Table[
   Flatten[{h} /.
      FindRoot[
       h == F[h,b,g,n], {h, -0.01}]][[1]]
   , {b, blist}, {g, glist}];

n = 100;
mlist = Range[-1. + 2/n, 1. - 2/n, 2./n];
m2list = mlist^2;
m3list = mlist^3;
logiotalist = Log[Binomial[n, n*(1 - mlist)/2]*2^(-n)];

d = 0.9;
glist = Range[0.4, 1, 0.01];
blist = Range[1.1, 10.1, 0.01];

ClearAll[F];
F[h_?NumericQ, b_, g_] :=
  Module[{var, cov, explist, μlist, mom1, mom2, mom3},
   explist = Exp[(b n h) mlist + (b n g/2) m2list + logiotalist + b n (-h + g/2)];
   μlist = explist/Total[explist];
   mom1 = μlist.mlist;
   mom2 = μlist.m2list;
   mom3 = μlist.m3list;
   var = Subtract[mom2, mom1 mom1];
   cov = Subtract[mom3, mom1 mom2];
   (-d b) (cov + 2. var)
   ];

{t1, r1} = F[0.1, blist[[1]], glist[[1]], n] // RepeatedTiming;
{t2, r2} = Fnew[0.1, blist[[1]], glist[[1]]] // RepeatedTiming;
Abs[r1 - r2]/r1
t1/t2

ParallelEvaluate[Off[General::munfl]];
heatdata = Developer`ToPackedArray[
     ParallelTable[
      Block[{h0, h},
       h0 = -0.01;
       Developer`ToPackedArray[
        Table[
         h0 = h /. FindRoot[h == Fnew[h, b, g], {h, h0}],
         {b, blist}]
        ]
       ],
      {g, glist}]
     ]; // AbsoluteTiming // First

explist = Exp[(b n h) mlist + (b n g/2) m2list + logiotalist + b n (-h + g/2)];